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",[568,1164,1166,1191],{"className":1165},[571],[568,1167,1169],{"className":1168},[575],[577,1170,1171],{"xmlns":579},[581,1172,1173,1188],{},[584,1174,1175],{},[1176,1177,1178,1180],"msup",{},[587,1179,712],{},[584,1181,1182,1184,1186],{},[587,1183,589],{},[591,1185,709],{},[730,1187,732],{},[597,1189,1190],{"encoding":599},"\\lambda^{n-1}",[568,1192,1194],{"className":1193,"ariaHidden":605},[604],[568,1195,1197,1201],{"className":1196},[609],[568,1198],{"className":1199,"style":1200},[613],"height:0.8141em;",[568,1202,1204,1207],{"className":1203},[618],[568,1205,712],{"className":1206},[618,619],[568,1208,1210],{"className":1209},[854],[568,1211,1213],{"className":1212},[858],[568,1214,1216],{"className":1215},[863],[568,1217,1219],{"className":1218,"style":1200},[867],[568,1220,1222,1225],{"style":1221},"top:-3.063em;margin-right:0.05em;",[568,1223],{"className":1224,"style":876},[875],[568,1226,1228],{"className":1227},[880,881,882,883],[568,1229,1231,1234,1237],{"className":1230},[618,883],[568,1232,589],{"className":1233},[618,619,883],[568,1235,709],{"className":1236},[628,883],[568,1238,732],{"className":1239},[618,883]," ist links ",[568,1242,1244,1285],{"className":1243},[571],[568,1245,1247],{"className":1246},[575],[577,1248,1249],{"xmlns":579},[581,1250,1251,1282],{},[584,1252,1253,1255,1257,1259,1271,1274,1276,1278,1280],{},[591,1254,703],{"stretchy":702},[591,1256,709],{},[730,1258,732],{},[1176,1260,1261,1263],{},[591,1262,718],{"stretchy":702},[584,1264,1265,1267,1269],{},[587,1266,589],{},[591,1268,709],{},[730,1270,732],{},[587,1272,1273],{"mathvariant":773},"Spur",[591,1275,699],{},[591,1277,703],{"stretchy":702},[587,1279,706],{},[591,1281,718],{"stretchy":702},[597,1283,1284],{"encoding":599},"(-1)^{n-1} \\operatorname{Spur}(A)",[568,1286,1288],{"className":1287,"ariaHidden":605},[604],[568,1289,1291,1295,1298,1301,1304,1342,1345,1352,1355,1358],{"className":1290},[609],[568,1292],{"className":1293,"style":1294},[613],"height:1.0641em;vertical-align:-0.25em;",[568,1296,703],{"className":1297},[795],[568,1299,709],{"className":1300},[618],[568,1302,732],{"className":1303},[618],[568,1305,1307,1310],{"className":1306},[824],[568,1308,718],{"className":1309},[824],[568,1311,1313],{"className":1312},[854],[568,1314,1316],{"className":1315},[858],[568,1317,1319],{"className":1318},[863],[568,1320,1322],{"className":1321,"style":1200},[867],[568,1323,1324,1327],{"style":1221},[568,1325],{"className":1326,"style":876},[875],[568,1328,1330],{"className":1329},[880,881,882,883],[568,1331,1333,1336,1339],{"className":1332},[618,883],[568,1334,589],{"className":1335},[618,619,883],[568,1337,709],{"className":1338},[628,883],[568,1340,732],{"className":1341},[618,883],[568,1343],{"className":1344,"style":989},[623],[568,1346,1348],{"className":1347},[791],[568,1349,1273],{"className":1350},[618,1351],"mathrm",[568,1353,703],{"className":1354},[795],[568,1356,706],{"className":1357},[618,619],[568,1359,718],{"className":1360},[824]," und rechts ",[568,1363,1365,1419],{"className":1364},[571],[568,1366,1368],{"className":1367},[575],[577,1369,1370],{"xmlns":579},[581,1371,1372,1416],{},[584,1373,1374,1376,1378,1380,1392,1394,1400,1403,1406,1408,1414],{},[591,1375,703],{"stretchy":702},[591,1377,709],{},[730,1379,732],{},[1176,1381,1382,1384],{},[591,1383,718],{"stretchy":702},[584,1385,1386,1388,1390],{},[587,1387,589],{},[591,1389,709],{},[730,1391,732],{},[591,1393,703],{"stretchy":702},[725,1395,1396,1398],{},[587,1397,712],{},[730,1399,732],{},[591,1401,1402],{},"+",[591,1404,1405],{},"…",[591,1407,1402],{},[725,1409,1410,1412],{},[587,1411,712],{},[587,1413,589],{},[591,1415,718],{"stretchy":702},[597,1417,1418],{"encoding":599},"(-1)^{n-1}(\\lambda_1 + \\ldots + \\lambda_n)",[568,1420,1422,1527,1545],{"className":1421,"ariaHidden":605},[604],[568,1423,1425,1428,1431,1434,1437,1475,1478,1518,1521,1524],{"className":1424},[609],[568,1426],{"className":1427,"style":1294},[613],[568,1429,703],{"className":1430},[795],[568,1432,709],{"className":1433},[618],[568,1435,732],{"className":1436},[618],[568,1438,1440,1443],{"className":1439},[824],[568,1441,718],{"className":1442},[824],[568,1444,1446],{"className":1445},[854],[568,1447,1449],{"className":1448},[858],[568,1450,1452],{"className":1451},[863],[568,1453,1455],{"className":1454,"style":1200},[867],[568,1456,1457,1460],{"style":1221},[568,1458],{"className":1459,"style":876},[875],[568,1461,1463],{"className":1462},[880,881,882,883],[568,1464,1466,1469,1472],{"className":1465},[618,883],[568,1467,589],{"className":1468},[618,619,883],[568,1470,709],{"className":1471},[628,883],[568,1473,732],{"className":1474},[618,883],[568,1476,703],{"className":1477},[795],[568,1479,1481,1484],{"className":1480},[618],[568,1482,712],{"className":1483},[618,619],[568,1485,1487],{"className":1486},[854],[568,1488,1490,1510],{"className":1489},[858,859],[568,1491,1493,1507],{"className":1492},[863],[568,1494,1496],{"className":1495,"style":868},[867],[568,1497,1498,1501],{"style":871},[568,1499],{"className":1500,"style":876},[875],[568,1502,1504],{"className":1503},[880,881,882,883],[568,1505,732],{"className":1506},[618,883],[568,1508,891],{"className":1509},[890],[568,1511,1513],{"className":1512},[863],[568,1514,1516],{"className":1515,"style":898},[867],[568,1517],{},[568,1519],{"className":1520,"style":624},[623],[568,1522,1402],{"className":1523},[628],[568,1525],{"className":1526,"style":624},[623],[568,1528,1530,1533,1536,1539,1542],{"className":1529},[609],[568,1531],{"className":1532,"style":614},[613],[568,1534,1405],{"className":1535},[993],[568,1537],{"className":1538,"style":624},[623],[568,1540,1402],{"className":1541},[628],[568,1543],{"className":1544,"style":624},[623],[568,1546,1548,1551,1591],{"className":1547},[609],[568,1549],{"className":1550,"style":787},[613],[568,1552,1554,1557],{"className":1553},[618],[568,1555,712],{"className":1556},[618,619],[568,1558,1560],{"className":1559},[854],[568,1561,1563,1583],{"className":1562},[858,859],[568,1564,1566,1580],{"className":1565},[863],[568,1567,1569],{"className":1568,"style":1018},[867],[568,1570,1571,1574],{"style":871},[568,1572],{"className":1573,"style":876},[875],[568,1575,1577],{"className":1576},[880,881,882,883],[568,1578,589],{"className":1579},[618,619,883],[568,1581,891],{"className":1582},[890],[568,1584,1586],{"className":1585},[863],[568,1587,1589],{"className":1588,"style":898},[867],[568,1590],{},[568,1592,718],{"className":1593},[824],[564,1595,1596],{},"Wichtig dabei:",[1598,1599,1600,1609],"ul",{},[1601,1602,1603,1604,1608],"li",{},"Jeder Eigenwert zählt so oft, wie seine ",[1605,1606,1607],"strong",{},"algebraische Vielfachheit"," angibt.",[1601,1610,1611,1612,1615],{},"Auch ",[1605,1613,1614],{},"komplexe"," Eigenwerte zählen mit. Bei reellen Matrizen treten sie in konjugierten Paaren auf, Summe und Produkt bleiben reell.",[564,1617,1618],{},"Folgerungen:",[1598,1620,1621,1682,1906],{},[1601,1622,1623,1652,1653,1681],{},[568,1624,1626,1639],{"className":1625},[571],[568,1627,1629],{"className":1628},[575],[577,1630,1631],{"xmlns":579},[581,1632,1633,1637],{},[584,1634,1635],{},[587,1636,706],{},[597,1638,706],{"encoding":599},[568,1640,1642],{"className":1641,"ariaHidden":605},[604],[568,1643,1645,1649],{"className":1644},[609],[568,1646],{"className":1647,"style":1648},[613],"height:0.6833em;",[568,1650,706],{"className":1651},[618,619]," ist genau dann invertierbar, wenn ",[568,1654,1656,1669],{"className":1655},[571],[568,1657,1659],{"className":1658},[575],[577,1660,1661],{"xmlns":579},[581,1662,1663,1667],{},[584,1664,1665],{},[730,1666,1086],{},[597,1668,1086],{"encoding":599},[568,1670,1672],{"className":1671,"ariaHidden":605},[604],[568,1673,1675,1678],{"className":1674},[609],[568,1676],{"className":1677,"style":1118},[613],[568,1679,1086],{"className":1680},[618]," kein Eigenwert ist.",[1601,1683,1684,1685,1737,1738,774],{},"Für ",[568,1686,1688,1706],{"className":1687},[571],[568,1689,1691],{"className":1690},[575],[577,1692,1693],{"xmlns":579},[581,1694,1695,1703],{},[584,1696,1697,1699,1701],{},[730,1698,747],{},[591,1700,593],{},[730,1702,747],{},[597,1704,1705],{"encoding":599},"2 \\times 2",[568,1707,1709,1728],{"className":1708,"ariaHidden":605},[604],[568,1710,1712,1716,1719,1722,1725],{"className":1711},[609],[568,1713],{"className":1714,"style":1715},[613],"height:0.7278em;vertical-align:-0.0833em;",[568,1717,747],{"className":1718},[618],[568,1720],{"className":1721,"style":624},[623],[568,1723,593],{"className":1724},[628],[568,1726],{"className":1727,"style":624},[623],[568,1729,1731,1734],{"className":1730},[609],[568,1732],{"className":1733,"style":1118},[613],[568,1735,747],{"className":1736},[618],"-Matrizen sind die Eigenwerte die Lösungen von ",[568,1739,1741,1789],{"className":1740},[571],[568,1742,1744],{"className":1743},[575],[577,1745,1746],{"xmlns":579},[581,1747,1748,1786],{},[584,1749,1750,1756,1758,1760,1762,1764,1766,1768,1772,1774,1776,1778,1780,1782,1784],{},[1176,1751,1752,1754],{},[587,1753,712],{},[730,1755,747],{},[591,1757,709],{},[587,1759,1273],{"mathvariant":773},[591,1761,699],{},[591,1763,703],{"stretchy":702},[587,1765,706],{},[591,1767,718],{"stretchy":702},[1769,1770,1771],"mtext",{}," ",[587,1773,712],{},[591,1775,1402],{},[587,1777,696],{},[591,1779,699],{},[587,1781,706],{},[591,1783,721],{},[730,1785,1086],{},[597,1787,1788],{"encoding":599},"\\lambda^2 - \\operatorname{Spur}(A)\\,\\lambda + \\det A = 0",[568,1790,1792,1837,1873,1897],{"className":1791,"ariaHidden":605},[604],[568,1793,1795,1799,1828,1831,1834],{"className":1794},[609],[568,1796],{"className":1797,"style":1798},[613],"height:0.8974em;vertical-align:-0.0833em;",[568,1800,1802,1805],{"className":1801},[618],[568,1803,712],{"className":1804},[618,619],[568,1806,1808],{"className":1807},[854],[568,1809,1811],{"className":1810},[858],[568,1812,1814],{"className":1813},[863],[568,1815,1817],{"className":1816,"style":1200},[867],[568,1818,1819,1822],{"style":1221},[568,1820],{"className":1821,"style":876},[875],[568,1823,1825],{"className":1824},[880,881,882,883],[568,1826,747],{"className":1827},[618,883],[568,1829],{"className":1830,"style":624},[623],[568,1832,709],{"className":1833},[628],[568,1835],{"className":1836,"style":624},[623],[568,1838,1840,1843,1849,1852,1855,1858,1861,1864,1867,1870],{"className":1839},[609],[568,1841],{"className":1842,"style":787},[613],[568,1844,1846],{"className":1845},[791],[568,1847,1273],{"className":1848},[618,1351],[568,1850,703],{"className":1851},[795],[568,1853,706],{"className":1854},[618,619],[568,1856,718],{"className":1857},[824],[568,1859],{"className":1860,"style":989},[623],[568,1862,712],{"className":1863},[618,619],[568,1865],{"className":1866,"style":624},[623],[568,1868,1402],{"className":1869},[628],[568,1871],{"className":1872,"style":624},[623],[568,1874,1876,1879,1882,1885,1888,1891,1894],{"className":1875},[609],[568,1877],{"className":1878,"style":1099},[613],[568,1880,696],{"className":1881},[791],[568,1883],{"className":1884,"style":989},[623],[568,1886,706],{"className":1887},[618,619],[568,1889],{"className":1890,"style":828},[623],[568,1892,721],{"className":1893},[832],[568,1895],{"className":1896,"style":828},[623],[568,1898,1900,1903],{"className":1899},[609],[568,1901],{"className":1902,"style":1118},[613],[568,1904,1086],{"className":1905},[618],[1601,1907,1908,1909,1912],{},"Die Formeln eignen sich gut als ",[1605,1910,1911],{},"Kontrolle"," nach einer Eigenwertberechnung.",[559,1914,1916],{"id":1915},"beispiele","Beispiele",[564,1918,1919,2327,2328,2456,2457,2485,2486,2485,2514,2542,2543,2747,2748,774],{},[568,1920,1922,2014],{"className":1921},[571],[568,1923,1925],{"className":1924},[575],[577,1926,1927],{"xmlns":579},[581,1928,1929,2011],{},[584,1930,1931,1933,1935],{},[587,1932,706],{},[591,1934,721],{},[584,1936,1937,1939,2009],{},[591,1938,703],{"fence":605},[1940,1941,1945,1968,1988],"mtable",{"rowspacing":1942,"columnalign":1943,"columnspacing":1944},"0.16em","center center center","1em",[1946,1947,1948,1956,1962],"mtr",{},[1949,1950,1951],"mtd",{},[1952,1953,1954],"mstyle",{"scriptlevel":1086,"displaystyle":702},[730,1955,747],{},[1949,1957,1958],{},[1952,1959,1960],{"scriptlevel":1086,"displaystyle":702},[730,1961,732],{},[1949,1963,1964],{},[1952,1965,1966],{"scriptlevel":1086,"displaystyle":702},[730,1967,1086],{},[1946,1969,1970,1976,1982],{},[1949,1971,1972],{},[1952,1973,1974],{"scriptlevel":1086,"displaystyle":702},[730,1975,732],{},[1949,1977,1978],{},[1952,1979,1980],{"scriptlevel":1086,"displaystyle":702},[730,1981,747],{},[1949,1983,1984],{},[1952,1985,1986],{"scriptlevel":1086,"displaystyle":702},[730,1987,1086],{},[1946,1989,1990,1996,2002],{},[1949,1991,1992],{},[1952,1993,1994],{"scriptlevel":1086,"displaystyle":702},[730,1995,1086],{},[1949,1997,1998],{},[1952,1999,2000],{"scriptlevel":1086,"displaystyle":702},[730,2001,1086],{},[1949,2003,2004],{},[1952,2005,2006],{"scriptlevel":1086,"displaystyle":702},[730,2007,2008],{},"3",[591,2010,718],{"fence":605},[597,2012,2013],{"encoding":599},"A = \\begin{pmatrix} 2 & 1 & 0 \\\\ 1 & 2 & 0 \\\\ 0 & 0 & 3 \\end{pmatrix}",[568,2015,2017,2035],{"className":2016,"ariaHidden":605},[604],[568,2018,2020,2023,2026,2029,2032],{"className":2019},[609],[568,2021],{"className":2022,"style":1648},[613],[568,2024,706],{"className":2025},[618,619],[568,2027],{"className":2028,"style":828},[623],[568,2030,721],{"className":2031},[832],[568,2033],{"className":2034,"style":828},[623],[568,2036,2038,2042],{"className":2037},[609],[568,2039],{"className":2040,"style":2041},[613],"height:3.6em;vertical-align:-1.55em;",[568,2043,2045,2096,2289],{"className":2044},[993],[568,2046,2048],{"className":2047},[795],[568,2049,2053],{"className":2050},[2051,2052],"delimsizing","mult",[568,2054,2056,2087],{"className":2055},[858,859],[568,2057,2059,2084],{"className":2058},[863],[568,2060,2063],{"className":2061,"style":2062},[867],"height:2.05em;",[568,2064,2066,2070],{"style":2065},"top:-4.05em;",[568,2067],{"className":2068,"style":2069},[875],"height:5.6em;",[568,2071,2073],{"style":2072},"width:0.875em;height:3.6em;",[2074,2075,2080],"svg",{"xmlns":2076,"width":2077,"height":2078,"viewBox":2079},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.875em","3.6em","0 0 875 3600",[2081,2082],"path",{"d":2083},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 l0,84c0.2,6,0,26,0,60c2,159.3,10,310.7,24,454c53.3,528,210,\n949.7,470,1265c4.7,6,9.7,11.7,15,17c0.7,0.7,7,1,19,1c0,0,18,0,18,0c4,-4,6,-7,6,-9\nc0,-2.7,-3.3,-8.7,-10,-18c-135.3,-192.7,-235.5,-414.3,-300.5,-665c-65,-250.7,-102.5,\n-544.7,-112.5,-882c-2,-104,-3,-167,-3,-189\nl0,-92c0,-162.7,5.7,-314,17,-454c20.7,-272,63.7,-513,129,-723c65.3,\n-210,155.3,-396.3,270,-559c6.7,-9.3,10,-15.3,10,-18z",[568,2085,891],{"className":2086},[890],[568,2088,2090],{"className":2089},[863],[568,2091,2094],{"className":2092,"style":2093},[867],"height:1.55em;",[568,2095],{},[568,2097,2099],{"className":2098},[618],[568,2100,2102,2163,2168,2171,2227,2230,2233],{"className":2101},[1940],[568,2103,2106],{"className":2104},[2105],"col-align-c",[568,2107,2109,2155],{"className":2108},[858,859],[568,2110,2112,2152],{"className":2111},[863],[568,2113,2115,2128,2140],{"className":2114,"style":2062},[867],[568,2116,2118,2122],{"style":2117},"top:-4.21em;",[568,2119],{"className":2120,"style":2121},[875],"height:3em;",[568,2123,2125],{"className":2124},[618],[568,2126,747],{"className":2127},[618],[568,2129,2131,2134],{"style":2130},"top:-3.01em;",[568,2132],{"className":2133,"style":2121},[875],[568,2135,2137],{"className":2136},[618],[568,2138,732],{"className":2139},[618],[568,2141,2143,2146],{"style":2142},"top:-1.81em;",[568,2144],{"className":2145,"style":2121},[875],[568,2147,2149],{"className":2148},[618],[568,2150,1086],{"className":2151},[618],[568,2153,891],{"className":2154},[890],[568,2156,2158],{"className":2157},[863],[568,2159,2161],{"className":2160,"style":2093},[867],[568,2162],{},[568,2164],{"className":2165,"style":2167},[2166],"arraycolsep","width:0.5em;",[568,2169],{"className":2170,"style":2167},[2166],[568,2172,2174],{"className":2173},[2105],[568,2175,2177,2219],{"className":2176},[858,859],[568,2178,2180,2216],{"className":2179},[863],[568,2181,2183,2194,2205],{"className":2182,"style":2062},[867],[568,2184,2185,2188],{"style":2117},[568,2186],{"className":2187,"style":2121},[875],[568,2189,2191],{"className":2190},[618],[568,2192,732],{"className":2193},[618],[568,2195,2196,2199],{"style":2130},[568,2197],{"className":2198,"style":2121},[875],[568,2200,2202],{"className":2201},[618],[568,2203,747],{"className":2204},[618],[568,2206,2207,2210],{"style":2142},[568,2208],{"className":2209,"style":2121},[875],[568,2211,2213],{"className":2212},[618],[568,2214,1086],{"className":2215},[618],[568,2217,891],{"className":2218},[890],[568,2220,2222],{"className":2221},[863],[568,2223,2225],{"className":2224,"style":2093},[867],[568,2226],{},[568,2228],{"className":2229,"style":2167},[2166],[568,2231],{"className":2232,"style":2167},[2166],[568,2234,2236],{"className":2235},[2105],[568,2237,2239,2281],{"className":2238},[858,859],[568,2240,2242,2278],{"className":2241},[863],[568,2243,2245,2256,2267],{"className":2244,"style":2062},[867],[568,2246,2247,2250],{"style":2117},[568,2248],{"className":2249,"style":2121},[875],[568,2251,2253],{"className":2252},[618],[568,2254,1086],{"className":2255},[618],[568,2257,2258,2261],{"style":2130},[568,2259],{"className":2260,"style":2121},[875],[568,2262,2264],{"className":2263},[618],[568,2265,1086],{"className":2266},[618],[568,2268,2269,2272],{"style":2142},[568,2270],{"className":2271,"style":2121},[875],[568,2273,2275],{"className":2274},[618],[568,2276,2008],{"className":2277},[618],[568,2279,891],{"className":2280},[890],[568,2282,2284],{"className":2283},[863],[568,2285,2287],{"className":2286,"style":2093},[867],[568,2288],{},[568,2290,2292],{"className":2291},[824],[568,2293,2295],{"className":2294},[2051,2052],[568,2296,2298,2319],{"className":2297},[858,859],[568,2299,2301,2316],{"className":2300},[863],[568,2302,2304],{"className":2303,"style":2062},[867],[568,2305,2306,2309],{"style":2065},[568,2307],{"className":2308,"style":2069},[875],[568,2310,2311],{"style":2072},[2074,2312,2313],{"xmlns":2076,"width":2077,"height":2078,"viewBox":2079},[2081,2314],{"d":2315},"M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,\n63,95c96.7,156.7,172.8,332.5,228.5,527.5c55.7,195,92.8,416.5,111.5,664.5\nc11.3,139.3,17,290.7,17,454c0,28,1.7,43,3.3,45l0,9\nc-3,4,-3.3,16.7,-3.3,38c0,162,-5.7,313.7,-17,455c-18.7,248,-55.8,469.3,-111.5,664\nc-55.7,194.7,-131.8,370.3,-228.5,527c-20.7,34.7,-41.7,66.3,-63,95c-2,3.3,-4,7,-6,11\nc0,7.3,5.7,11,17,11c0,0,11,0,11,0c9.3,0,14.3,-0.3,15,-1c5.3,-5.3,10.3,-11,15,-17\nc242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558\nl0,-144c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,\n-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z",[568,2317,891],{"className":2318},[890],[568,2320,2322],{"className":2321},[863],[568,2323,2325],{"className":2324,"style":2093},[867],[568,2326],{}," hat das charakteristische Polynom ",[568,2329,2331,2367],{"className":2330},[571],[568,2332,2334],{"className":2333},[575],[577,2335,2336],{"xmlns":579},[581,2337,2338,2364],{},[584,2339,2340,2342,2344,2346,2348,2350,2352,2354,2356,2358],{},[591,2341,703],{"stretchy":702},[730,2343,732],{},[591,2345,709],{},[587,2347,712],{},[591,2349,718],{"stretchy":702},[591,2351,703],{"stretchy":702},[730,2353,2008],{},[591,2355,709],{},[587,2357,712],{},[1176,2359,2360,2362],{},[591,2361,718],{"stretchy":702},[730,2363,747],{},[597,2365,2366],{"encoding":599},"(1 - \\lambda)(3 - \\lambda)^2",[568,2368,2370,2391,2418],{"className":2369,"ariaHidden":605},[604],[568,2371,2373,2376,2379,2382,2385,2388],{"className":2372},[609],[568,2374],{"className":2375,"style":787},[613],[568,2377,703],{"className":2378},[795],[568,2380,732],{"className":2381},[618],[568,2383],{"className":2384,"style":624},[623],[568,2386,709],{"className":2387},[628],[568,2389],{"className":2390,"style":624},[623],[568,2392,2394,2397,2400,2403,2406,2409,2412,2415],{"className":2393},[609],[568,2395],{"className":2396,"style":787},[613],[568,2398,712],{"className":2399},[618,619],[568,2401,718],{"className":2402},[824],[568,2404,703],{"className":2405},[795],[568,2407,2008],{"className":2408},[618],[568,2410],{"className":2411,"style":624},[623],[568,2413,709],{"className":2414},[628],[568,2416],{"className":2417,"style":624},[623],[568,2419,2421,2424,2427],{"className":2420},[609],[568,2422],{"className":2423,"style":1294},[613],[568,2425,712],{"className":2426},[618,619],[568,2428,2430,2433],{"className":2429},[824],[568,2431,718],{"className":2432},[824],[568,2434,2436],{"className":2435},[854],[568,2437,2439],{"className":2438},[858],[568,2440,2442],{"className":2441},[863],[568,2443,2445],{"className":2444,"style":1200},[867],[568,2446,2447,2450],{"style":1221},[568,2448],{"className":2449,"style":876},[875],[568,2451,2453],{"className":2452},[880,881,882,883],[568,2454,747],{"className":2455},[618,883],", also die Eigenwerte ",[568,2458,2460,2473],{"className":2459},[571],[568,2461,2463],{"className":2462},[575],[577,2464,2465],{"xmlns":579},[581,2466,2467,2471],{},[584,2468,2469],{},[730,2470,732],{},[597,2472,732],{"encoding":599},[568,2474,2476],{"className":2475,"ariaHidden":605},[604],[568,2477,2479,2482],{"className":2478},[609],[568,2480],{"className":2481,"style":1118},[613],[568,2483,732],{"className":2484},[618],", ",[568,2487,2489,2502],{"className":2488},[571],[568,2490,2492],{"className":2491},[575],[577,2493,2494],{"xmlns":579},[581,2495,2496,2500],{},[584,2497,2498],{},[730,2499,2008],{},[597,2501,2008],{"encoding":599},[568,2503,2505],{"className":2504,"ariaHidden":605},[604],[568,2506,2508,2511],{"className":2507},[609],[568,2509],{"className":2510,"style":1118},[613],[568,2512,2008],{"className":2513},[618],[568,2515,2517,2530],{"className":2516},[571],[568,2518,2520],{"className":2519},[575],[577,2521,2522],{"xmlns":579},[581,2523,2524,2528],{},[584,2525,2526],{},[730,2527,2008],{},[597,2529,2008],{"encoding":599},[568,2531,2533],{"className":2532,"ariaHidden":605},[604],[568,2534,2536,2539],{"className":2535},[609],[568,2537],{"className":2538,"style":1118},[613],[568,2540,2008],{"className":2541},[618],". Kontrolle: ",[568,2544,2546,2597],{"className":2545},[571],[568,2547,2549],{"className":2548},[575],[577,2550,2551],{"xmlns":579},[581,2552,2553,2594],{},[584,2554,2555,2557,2559,2561,2563,2565,2567,2569,2571,2573,2575,2577,2579,2582,2584,2586,2588,2590,2592],{},[587,2556,1273],{"mathvariant":773},[591,2558,699],{},[591,2560,703],{"stretchy":702},[587,2562,706],{},[591,2564,718],{"stretchy":702},[591,2566,721],{},[730,2568,747],{},[591,2570,1402],{},[730,2572,747],{},[591,2574,1402],{},[730,2576,2008],{},[591,2578,721],{},[730,2580,2581],{},"7",[591,2583,721],{},[730,2585,732],{},[591,2587,1402],{},[730,2589,2008],{},[591,2591,1402],{},[730,2593,2008],{},[597,2595,2596],{"encoding":599},"\\operatorname{Spur}(A) = 2 + 2 + 3 = 7 = 1 + 3 + 3",[568,2598,2600,2630,2648,2666,2684,2702,2720,2738],{"className":2599,"ariaHidden":605},[604],[568,2601,2603,2606,2612,2615,2618,2621,2624,2627],{"className":2602},[609],[568,2604],{"className":2605,"style":787},[613],[568,2607,2609],{"className":2608},[791],[568,2610,1273],{"className":2611},[618,1351],[568,2613,703],{"className":2614},[795],[568,2616,706],{"className":2617},[618,619],[568,2619,718],{"className":2620},[824],[568,2622],{"className":2623,"style":828},[623],[568,2625,721],{"className":2626},[832],[568,2628],{"className":2629,"style":828},[623],[568,2631,2633,2636,2639,2642,2645],{"className":2632},[609],[568,2634],{"className":2635,"style":1715},[613],[568,2637,747],{"className":2638},[618],[568,2640],{"className":2641,"style":624},[623],[568,2643,1402],{"className":2644},[628],[568,2646],{"className":2647,"style":624},[623],[568,2649,2651,2654,2657,2660,2663],{"className":2650},[609],[568,2652],{"className":2653,"style":1715},[613],[568,2655,747],{"className":2656},[618],[568,2658],{"className":2659,"style":624},[623],[568,2661,1402],{"className":2662},[628],[568,2664],{"className":2665,"style":624},[623],[568,2667,2669,2672,2675,2678,2681],{"className":2668},[609],[568,2670],{"className":2671,"style":1118},[613],[568,2673,2008],{"className":2674},[618],[568,2676],{"className":2677,"style":828},[623],[568,2679,721],{"className":2680},[832],[568,2682],{"className":2683,"style":828},[623],[568,2685,2687,2690,2693,2696,2699],{"className":2686},[609],[568,2688],{"className":2689,"style":1118},[613],[568,2691,2581],{"className":2692},[618],[568,2694],{"className":2695,"style":828},[623],[568,2697,721],{"className":2698},[832],[568,2700],{"className":2701,"style":828},[623],[568,2703,2705,2708,2711,2714,2717],{"className":2704},[609],[568,2706],{"className":2707,"style":1715},[613],[568,2709,732],{"className":2710},[618],[568,2712],{"className":2713,"style":624},[623],[568,2715,1402],{"className":2716},[628],[568,2718],{"className":2719,"style":624},[623],[568,2721,2723,2726,2729,2732,2735],{"className":2722},[609],[568,2724],{"className":2725,"style":1715},[613],[568,2727,2008],{"className":2728},[618],[568,2730],{"className":2731,"style":624},[623],[568,2733,1402],{"className":2734},[628],[568,2736],{"className":2737,"style":624},[623],[568,2739,2741,2744],{"className":2740},[609],[568,2742],{"className":2743,"style":1118},[613],[568,2745,2008],{"className":2746},[618]," und ",[568,2749,2751,2804],{"className":2750},[571],[568,2752,2754],{"className":2753},[575],[577,2755,2756],{"xmlns":579},[581,2757,2758,2801],{},[584,2759,2760,2762,2764,2766,2768,2770,2773,2775,2778,2780,2782,2784,2786,2789,2791,2793,2795,2797,2799],{},[587,2761,696],{},[591,2763,699],{},[587,2765,706],{},[591,2767,721],{},[730,2769,2008],{},[591,2771,2772],{},"⋅",[591,2774,703],{"stretchy":702},[730,2776,2777],{},"4",[591,2779,709],{},[730,2781,732],{},[591,2783,718],{"stretchy":702},[591,2785,721],{},[730,2787,2788],{},"9",[591,2790,721],{},[730,2792,732],{},[591,2794,2772],{},[730,2796,2008],{},[591,2798,2772],{},[730,2800,2008],{},[597,2802,2803],{"encoding":599},"\\det A = 3 \\cdot (4 - 1) = 9 = 1 \\cdot 3 \\cdot 3",[568,2805,2807,2831,2849,2870,2891,2909,2927,2945],{"className":2806,"ariaHidden":605},[604],[568,2808,2810,2813,2816,2819,2822,2825,2828],{"className":2809},[609],[568,2811],{"className":2812,"style":1099},[613],[568,2814,696],{"className":2815},[791],[568,2817],{"className":2818,"style":989},[623],[568,2820,706],{"className":2821},[618,619],[568,2823],{"className":2824,"style":828},[623],[568,2826,721],{"className":2827},[832],[568,2829],{"className":2830,"style":828},[623],[568,2832,2834,2837,2840,2843,2846],{"className":2833},[609],[568,2835],{"className":2836,"style":1118},[613],[568,2838,2008],{"className":2839},[618],[568,2841],{"className":2842,"style":624},[623],[568,2844,2772],{"className":2845},[628],[568,2847],{"className":2848,"style":624},[623],[568,2850,2852,2855,2858,2861,2864,2867],{"className":2851},[609],[568,2853],{"className":2854,"style":787},[613],[568,2856,703],{"className":2857},[795],[568,2859,2777],{"className":2860},[618],[568,2862],{"className":2863,"style":624},[623],[568,2865,709],{"className":2866},[628],[568,2868],{"className":2869,"style":624},[623],[568,2871,2873,2876,2879,2882,2885,2888],{"className":2872},[609],[568,2874],{"className":2875,"style":787},[613],[568,2877,732],{"className":2878},[618],[568,2880,718],{"className":2881},[824],[568,2883],{"className":2884,"style":828},[623],[568,2886,721],{"className":2887},[832],[568,2889],{"className":2890,"style":828},[623],[568,2892,2894,2897,2900,2903,2906],{"className":2893},[609],[568,2895],{"className":2896,"style":1118},[613],[568,2898,2788],{"className":2899},[618],[568,2901],{"className":2902,"style":828},[623],[568,2904,721],{"className":2905},[832],[568,2907],{"className":2908,"style":828},[623],[568,2910,2912,2915,2918,2921,2924],{"className":2911},[609],[568,2913],{"className":2914,"style":1118},[613],[568,2916,732],{"className":2917},[618],[568,2919],{"className":2920,"style":624},[623],[568,2922,2772],{"className":2923},[628],[568,2925],{"className":2926,"style":624},[623],[568,2928,2930,2933,2936,2939,2942],{"className":2929},[609],[568,2931],{"className":2932,"style":1118},[613],[568,2934,2008],{"className":2935},[618],[568,2937],{"className":2938,"style":624},[623],[568,2940,2772],{"className":2941},[628],[568,2943],{"className":2944,"style":624},[623],[568,2946,2948,2951],{"className":2947},[609],[568,2949],{"className":2950,"style":1118},[613],[568,2952,2008],{"className":2953},[618],[564,2955,2956,2957,3199,3200,3301,3302,3385,3386,3434,3435,3596],{},"Die Drehmatrix ",[568,2958,2960,3036],{"className":2959},[571],[568,2961,2963],{"className":2962},[575],[577,2964,2965],{"xmlns":579},[581,2966,2967,3033],{},[584,2968,2969,2971,3031],{},[591,2970,703],{"fence":605},[1940,2972,2974,3005],{"rowspacing":1942,"columnalign":2973,"columnspacing":1944},"center center",[1946,2975,2976,2990],{},[1949,2977,2978],{},[1952,2979,2980],{"scriptlevel":1086,"displaystyle":702},[584,2981,2982,2985,2987],{},[587,2983,2984],{},"cos",[591,2986,699],{},[587,2988,2989],{},"φ",[1949,2991,2992],{},[1952,2993,2994],{"scriptlevel":1086,"displaystyle":702},[584,2995,2996,2998,3001,3003],{},[591,2997,709],{},[587,2999,3000],{},"sin",[591,3002,699],{},[587,3004,2989],{},[1946,3006,3007,3019],{},[1949,3008,3009],{},[1952,3010,3011],{"scriptlevel":1086,"displaystyle":702},[584,3012,3013,3015,3017],{},[587,3014,3000],{},[591,3016,699],{},[587,3018,2989],{},[1949,3020,3021],{},[1952,3022,3023],{"scriptlevel":1086,"displaystyle":702},[584,3024,3025,3027,3029],{},[587,3026,2984],{},[591,3028,699],{},[587,3030,2989],{},[591,3032,718],{"fence":605},[597,3034,3035],{"encoding":599},"\\begin{pmatrix} \\cos\\varphi & -\\sin\\varphi \\\\ \\sin\\varphi & \\cos\\varphi \\end{pmatrix}",[568,3037,3039],{"className":3038,"ariaHidden":605},[604],[568,3040,3042,3046],{"className":3041},[609],[568,3043],{"className":3044,"style":3045},[613],"height:2.4em;vertical-align:-0.95em;",[568,3047,3049,3057,3193],{"className":3048},[993],[568,3050,3054],{"className":3051,"style":3053},[795,3052],"delimcenter","top:0em;",[568,3055,703],{"className":3056},[2051,882],[568,3058,3060],{"className":3059},[618],[568,3061,3063,3124,3127,3130],{"className":3062},[1940],[568,3064,3066],{"className":3065},[2105],[568,3067,3069,3115],{"className":3068},[858,859],[568,3070,3072,3112],{"className":3071},[863],[568,3073,3076,3094],{"className":3074,"style":3075},[867],"height:1.45em;",[568,3077,3079,3082],{"style":3078},"top:-3.61em;",[568,3080],{"className":3081,"style":2121},[875],[568,3083,3085,3088,3091],{"className":3084},[618],[568,3086,2984],{"className":3087},[791],[568,3089],{"className":3090,"style":989},[623],[568,3092,2989],{"className":3093},[618,619],[568,3095,3097,3100],{"style":3096},"top:-2.41em;",[568,3098],{"className":3099,"style":2121},[875],[568,3101,3103,3106,3109],{"className":3102},[618],[568,3104,3000],{"className":3105},[791],[568,3107],{"className":3108,"style":989},[623],[568,3110,2989],{"className":3111},[618,619],[568,3113,891],{"className":3114},[890],[568,3116,3118],{"className":3117},[863],[568,3119,3122],{"className":3120,"style":3121},[867],"height:0.95em;",[568,3123],{},[568,3125],{"className":3126,"style":2167},[2166],[568,3128],{"className":3129,"style":2167},[2166],[568,3131,3133],{"className":3132},[2105],[568,3134,3136,3185],{"className":3135},[858,859],[568,3137,3139,3182],{"className":3138},[863],[568,3140,3142,3165],{"className":3141,"style":3075},[867],[568,3143,3144,3147],{"style":3078},[568,3145],{"className":3146,"style":2121},[875],[568,3148,3150,3153,3156,3159,3162],{"className":3149},[618],[568,3151,709],{"className":3152},[618],[568,3154],{"className":3155,"style":989},[623],[568,3157,3000],{"className":3158},[791],[568,3160],{"className":3161,"style":989},[623],[568,3163,2989],{"className":3164},[618,619],[568,3166,3167,3170],{"style":3096},[568,3168],{"className":3169,"style":2121},[875],[568,3171,3173,3176,3179],{"className":3172},[618],[568,3174,2984],{"className":3175},[791],[568,3177],{"className":3178,"style":989},[623],[568,3180,2989],{"className":3181},[618,619],[568,3183,891],{"className":3184},[890],[568,3186,3188],{"className":3187},[863],[568,3189,3191],{"className":3190,"style":3121},[867],[568,3192],{},[568,3194,3196],{"className":3195,"style":3053},[824,3052],[568,3197,718],{"className":3198},[2051,882]," hat für ",[568,3201,3203,3226],{"className":3202},[571],[568,3204,3206],{"className":3205},[575],[577,3207,3208],{"xmlns":579},[581,3209,3210,3223],{},[584,3211,3212,3214,3217,3220],{},[587,3213,2989],{},[591,3215,3216],{"mathvariant":773},"∉",[587,3218,3219],{},"π",[587,3221,3222],{"mathvariant":657},"Z",[597,3224,3225],{"encoding":599},"\\varphi \\notin \\pi\\mathbb{Z}",[568,3227,3229,3288],{"className":3228,"ariaHidden":605},[604],[568,3230,3232,3235,3238,3241,3285],{"className":3231},[609],[568,3233],{"className":3234,"style":787},[613],[568,3236,2989],{"className":3237},[618,619],[568,3239],{"className":3240,"style":828},[623],[568,3242,3244,3251],{"className":3243},[832],[568,3245,3247],{"className":3246},[618],[568,3248,3250],{"className":3249},[832],"∈",[568,3252,3255],{"className":3253},[618,3254],"vbox",[568,3256,3259],{"className":3257},[3258],"thinbox",[568,3260,3263,3266,3281],{"className":3261},[3262],"llap",[568,3264],{"className":3265,"style":787},[613],[568,3267,3270],{"className":3268},[3269],"inner",[568,3271,3273,3277],{"className":3272},[618],[568,3274,3276],{"className":3275},[618],"\u002F",[568,3278],{"className":3279,"style":3280},[623],"margin-right:0.0556em;",[568,3282],{"className":3283},[3284],"fix",[568,3286],{"className":3287,"style":828},[623],[568,3289,3291,3294,3298],{"className":3290},[609],[568,3292],{"className":3293,"style":671},[613],[568,3295,3219],{"className":3296,"style":3297},[618,619],"margin-right:0.0359em;",[568,3299,3222],{"className":3300},[618,675]," keine reellen Eigenwerte, sondern ",[568,3303,3305,3335],{"className":3304},[571],[568,3306,3308],{"className":3307},[575],[577,3309,3310],{"xmlns":579},[581,3311,3312,3332],{},[584,3313,3314,3316,3318,3320,3323,3326,3328,3330],{},[587,3315,2984],{},[591,3317,699],{},[587,3319,2989],{},[591,3321,3322],{},"±",[587,3324,3325],{"mathvariant":773},"i",[587,3327,3000],{},[591,3329,699],{},[587,3331,2989],{},[597,3333,3334],{"encoding":599},"\\cos\\varphi \\pm \\mathrm{i}\\sin\\varphi",[568,3336,3338,3363],{"className":3337,"ariaHidden":605},[604],[568,3339,3341,3345,3348,3351,3354,3357,3360],{"className":3340},[609],[568,3342],{"className":3343,"style":3344},[613],"height:0.7778em;vertical-align:-0.1944em;",[568,3346,2984],{"className":3347},[791],[568,3349],{"className":3350,"style":989},[623],[568,3352,2989],{"className":3353},[618,619],[568,3355],{"className":3356,"style":624},[623],[568,3358,3322],{"className":3359},[628],[568,3361],{"className":3362,"style":624},[623],[568,3364,3366,3370,3373,3376,3379,3382],{"className":3365},[609],[568,3367],{"className":3368,"style":3369},[613],"height:0.8623em;vertical-align:-0.1944em;",[568,3371,3325],{"className":3372},[618,1351],[568,3374],{"className":3375,"style":989},[623],[568,3377,3000],{"className":3378},[791],[568,3380],{"className":3381,"style":989},[623],[568,3383,2989],{"className":3384},[618,619],". Ihre Summe ist ",[568,3387,3389,3409],{"className":3388},[571],[568,3390,3392],{"className":3391},[575],[577,3393,3394],{"xmlns":579},[581,3395,3396,3406],{},[584,3397,3398,3400,3402,3404],{},[730,3399,747],{},[587,3401,2984],{},[591,3403,699],{},[587,3405,2989],{},[597,3407,3408],{"encoding":599},"2\\cos\\varphi",[568,3410,3412],{"className":3411,"ariaHidden":605},[604],[568,3413,3415,3419,3422,3425,3428,3431],{"className":3414},[609],[568,3416],{"className":3417,"style":3418},[613],"height:0.8389em;vertical-align:-0.1944em;",[568,3420,747],{"className":3421},[618],[568,3423],{"className":3424,"style":989},[623],[568,3426,2984],{"className":3427},[791],[568,3429],{"className":3430,"style":989},[623],[568,3432,2989],{"className":3433},[618,619]," (die Spur), ihr Produkt ",[568,3436,3438,3480],{"className":3437},[571],[568,3439,3441],{"className":3440},[575],[577,3442,3443],{"xmlns":579},[581,3444,3445,3477],{},[584,3446,3447,3457,3459,3461,3471,3473,3475],{},[1176,3448,3449,3455],{},[584,3450,3451,3453],{},[587,3452,2984],{},[591,3454,699],{},[730,3456,747],{},[587,3458,2989],{},[591,3460,1402],{},[1176,3462,3463,3469],{},[584,3464,3465,3467],{},[587,3466,3000],{},[591,3468,699],{},[730,3470,747],{},[587,3472,2989],{},[591,3474,721],{},[730,3476,732],{},[597,3478,3479],{"encoding":599},"\\cos^2\\varphi + \\sin^2\\varphi = 1",[568,3481,3483,3534,3587],{"className":3482,"ariaHidden":605},[604],[568,3484,3486,3490,3519,3522,3525,3528,3531],{"className":3485},[609],[568,3487],{"className":3488,"style":3489},[613],"height:1.0085em;vertical-align:-0.1944em;",[568,3491,3493,3496],{"className":3492},[791],[568,3494,2984],{"className":3495},[791],[568,3497,3499],{"className":3498},[854],[568,3500,3502],{"className":3501},[858],[568,3503,3505],{"className":3504},[863],[568,3506,3508],{"className":3507,"style":1200},[867],[568,3509,3510,3513],{"style":1221},[568,3511],{"className":3512,"style":876},[875],[568,3514,3516],{"className":3515},[880,881,882,883],[568,3517,747],{"className":3518},[618,883],[568,3520],{"className":3521,"style":989},[623],[568,3523,2989],{"className":3524},[618,619],[568,3526],{"className":3527,"style":624},[623],[568,3529,1402],{"className":3530},[628],[568,3532],{"className":3533,"style":624},[623],[568,3535,3537,3541,3572,3575,3578,3581,3584],{"className":3536},[609],[568,3538],{"className":3539,"style":3540},[613],"height:1.0663em;vertical-align:-0.1944em;",[568,3542,3544,3547],{"className":3543},[791],[568,3545,3000],{"className":3546},[791],[568,3548,3550],{"className":3549},[854],[568,3551,3553],{"className":3552},[858],[568,3554,3556],{"className":3555},[863],[568,3557,3560],{"className":3558,"style":3559},[867],"height:0.8719em;",[568,3561,3563,3566],{"style":3562},"top:-3.1208em;margin-right:0.05em;",[568,3564],{"className":3565,"style":876},[875],[568,3567,3569],{"className":3568},[880,881,882,883],[568,3570,747],{"className":3571},[618,883],[568,3573],{"className":3574,"style":989},[623],[568,3576,2989],{"className":3577},[618,619],[568,3579],{"className":3580,"style":828},[623],[568,3582,721],{"className":3583},[832],[568,3585],{"className":3586,"style":828},[623],[568,3588,3590,3593],{"className":3589},[609],[568,3591],{"className":3592,"style":1118},[613],[568,3594,732],{"className":3595},[618]," (die Determinante).",[564,3598,3599,3600,3650,3651,3680,3681,3710,3711,3739,3740,3813,3814,774],{},"Eine ",[568,3601,3603,3620],{"className":3602},[571],[568,3604,3606],{"className":3605},[575],[577,3607,3608],{"xmlns":579},[581,3609,3610,3618],{},[584,3611,3612,3614,3616],{},[730,3613,747],{},[591,3615,593],{},[730,3617,747],{},[597,3619,1705],{"encoding":599},[568,3621,3623,3641],{"className":3622,"ariaHidden":605},[604],[568,3624,3626,3629,3632,3635,3638],{"className":3625},[609],[568,3627],{"className":3628,"style":1715},[613],[568,3630,747],{"className":3631},[618],[568,3633],{"className":3634,"style":624},[623],[568,3636,593],{"className":3637},[628],[568,3639],{"className":3640,"style":624},[623],[568,3642,3644,3647],{"className":3643},[609],[568,3645],{"className":3646,"style":1118},[613],[568,3648,747],{"className":3649},[618],"-Matrix mit Spur ",[568,3652,3654,3668],{"className":3653},[571],[568,3655,3657],{"className":3656},[575],[577,3658,3659],{"xmlns":579},[581,3660,3661,3666],{},[584,3662,3663],{},[730,3664,3665],{},"6",[597,3667,3665],{"encoding":599},[568,3669,3671],{"className":3670,"ariaHidden":605},[604],[568,3672,3674,3677],{"className":3673},[609],[568,3675],{"className":3676,"style":1118},[613],[568,3678,3665],{"className":3679},[618],", Determinante ",[568,3682,3684,3698],{"className":3683},[571],[568,3685,3687],{"className":3686},[575],[577,3688,3689],{"xmlns":579},[581,3690,3691,3696],{},[584,3692,3693],{},[730,3694,3695],{},"8",[597,3697,3695],{"encoding":599},[568,3699,3701],{"className":3700,"ariaHidden":605},[604],[568,3702,3704,3707],{"className":3703},[609],[568,3705],{"className":3706,"style":1118},[613],[568,3708,3695],{"className":3709},[618]," und einem bekannten Eigenwert ",[568,3712,3714,3727],{"className":3713},[571],[568,3715,3717],{"className":3716},[575],[577,3718,3719],{"xmlns":579},[581,3720,3721,3725],{},[584,3722,3723],{},[730,3724,747],{},[597,3726,747],{"encoding":599},[568,3728,3730],{"className":3729,"ariaHidden":605},[604],[568,3731,3733,3736],{"className":3732},[609],[568,3734],{"className":3735,"style":1118},[613],[568,3737,747],{"className":3738},[618]," hat als zweiten Eigenwert ",[568,3741,3743,3765],{"className":3742},[571],[568,3744,3746],{"className":3745},[575],[577,3747,3748],{"xmlns":579},[581,3749,3750,3762],{},[584,3751,3752,3754,3756,3758,3760],{},[730,3753,3665],{},[591,3755,709],{},[730,3757,747],{},[591,3759,721],{},[730,3761,2777],{},[597,3763,3764],{"encoding":599},"6 - 2 = 4",[568,3766,3768,3786,3804],{"className":3767,"ariaHidden":605},[604],[568,3769,3771,3774,3777,3780,3783],{"className":3770},[609],[568,3772],{"className":3773,"style":1715},[613],[568,3775,3665],{"className":3776},[618],[568,3778],{"className":3779,"style":624},[623],[568,3781,709],{"className":3782},[628],[568,3784],{"className":3785,"style":624},[623],[568,3787,3789,3792,3795,3798,3801],{"className":3788},[609],[568,3790],{"className":3791,"style":1118},[613],[568,3793,747],{"className":3794},[618],[568,3796],{"className":3797,"style":828},[623],[568,3799,721],{"className":3800},[832],[568,3802],{"className":3803,"style":828},[623],[568,3805,3807,3810],{"className":3806},[609],[568,3808],{"className":3809,"style":1118},[613],[568,3811,2777],{"className":3812},[618],". Probe: ",[568,3815,3817,3839],{"className":3816},[571],[568,3818,3820],{"className":3819},[575],[577,3821,3822],{"xmlns":579},[581,3823,3824,3836],{},[584,3825,3826,3828,3830,3832,3834],{},[730,3827,747],{},[591,3829,2772],{},[730,3831,2777],{},[591,3833,721],{},[730,3835,3695],{},[597,3837,3838],{"encoding":599},"2 \\cdot 4 = 8",[568,3840,3842,3860,3878],{"className":3841,"ariaHidden":605},[604],[568,3843,3845,3848,3851,3854,3857],{"className":3844},[609],[568,3846],{"className":3847,"style":1118},[613],[568,3849,747],{"className":3850},[618],[568,3852],{"className":3853,"style":624},[623],[568,3855,2772],{"className":3856},[628],[568,3858],{"className":3859,"style":624},[623],[568,3861,3863,3866,3869,3872,3875],{"className":3862},[609],[568,3864],{"className":3865,"style":1118},[613],[568,3867,2777],{"className":3868},[618],[568,3870],{"className":3871,"style":828},[623],[568,3873,721],{"className":3874},[832],[568,3876],{"className":3877,"style":828},[623],[568,3879,3881,3884],{"className":3880},[609],[568,3882],{"className":3883,"style":1118},[613],[568,3885,3695],{"className":3886},[618],[559,3888,3890],{"id":3889},"typische-fehler","Typische Fehler",[1598,3892,3893,3998,4001],{},[1601,3894,3895,3896,3969,3970,774],{},"Mehrfache Eigenwerte nur einmal zählen. Im ersten Beispiel ergäbe ",[568,3897,3899,3921],{"className":3898},[571],[568,3900,3902],{"className":3901},[575],[577,3903,3904],{"xmlns":579},[581,3905,3906,3918],{},[584,3907,3908,3910,3912,3914,3916],{},[730,3909,732],{},[591,3911,1402],{},[730,3913,2008],{},[591,3915,721],{},[730,3917,2777],{},[597,3919,3920],{"encoding":599},"1 + 3 = 4",[568,3922,3924,3942,3960],{"className":3923,"ariaHidden":605},[604],[568,3925,3927,3930,3933,3936,3939],{"className":3926},[609],[568,3928],{"className":3929,"style":1715},[613],[568,3931,732],{"className":3932},[618],[568,3934],{"className":3935,"style":624},[623],[568,3937,1402],{"className":3938},[628],[568,3940],{"className":3941,"style":624},[623],[568,3943,3945,3948,3951,3954,3957],{"className":3944},[609],[568,3946],{"className":3947,"style":1118},[613],[568,3949,2008],{"className":3950},[618],[568,3952],{"className":3953,"style":828},[623],[568,3955,721],{"className":3956},[832],[568,3958],{"className":3959,"style":828},[623],[568,3961,3963,3966],{"className":3962},[609],[568,3964],{"className":3965,"style":1118},[613],[568,3967,2777],{"className":3968},[618]," nicht die Spur ",[568,3971,3973,3986],{"className":3972},[571],[568,3974,3976],{"className":3975},[575],[577,3977,3978],{"xmlns":579},[581,3979,3980,3984],{},[584,3981,3982],{},[730,3983,2581],{},[597,3985,2581],{"encoding":599},[568,3987,3989],{"className":3988,"ariaHidden":605},[604],[568,3990,3992,3995],{"className":3991},[609],[568,3993],{"className":3994,"style":1118},[613],[568,3996,2581],{"className":3997},[618],[1601,3999,4000],{},"Komplexe Eigenwerte weglassen und dann einen Widerspruch zur Spur finden.",[1601,4002,4003],{},"Die Diagonaleinträge für die Eigenwerte halten. Das stimmt nur bei Dreiecks- und Diagonalmatrizen; die Summe stimmt aber immer.",{"title":4005,"searchDepth":4006,"depth":4006,"links":4007},"",2,[4008,4009,4010],{"id":561,"depth":4006,"text":562},{"id":1915,"depth":4006,"text":1916},{"id":3889,"depth":4006,"text":3890},[4012,4015,4018,4021,4024,4027,4030,4033,4035,4037],{"lhs":4013,"rhs":4014},"((a + d) + sqrt((a - d)^2 + 4*b*c))\u002F2 + ((a + d) - sqrt((a - d)^2 + 4*b*c))\u002F2","a + d",{"lhs":4016,"rhs":4017},"((a + d) + sqrt((a - d)^2 + 4*b*c))\u002F2 * ((a + d) - sqrt((a - d)^2 + 4*b*c))\u002F2","a*d - b*c",{"lhs":4019,"rhs":4020},"det([[2 - x, 1, 0], [1, 2 - x, 0], [0, 0, 3 - x]])","(1 - x)*(3 - x)^2",{"lhs":4022,"rhs":4023},"trace([[2, 1, 0], [1, 2, 0], [0, 0, 3]])","1 + 3 + 3",{"lhs":4025,"rhs":4026},"det([[2, 1, 0], [1, 2, 0], [0, 0, 3]])","1 * 3 * 3",{"lhs":4028,"rhs":4029},"2*cos(x)","trace([[cos(x), -sin(x)], [sin(x), cos(x)]])",{"lhs":4031,"rhs":4032},"cos(x)^2 + sin(x)^2","det([[cos(x), -sin(x)], [sin(x), cos(x)]])",{"lhs":4034,"rhs":732},"re((cos(x) + i*sin(x)) * (cos(x) - i*sin(x)))",{"lhs":4036,"rhs":2777},"6 - 2",{"lhs":4038,"rhs":3695},"2 * 4","A \\in \\mathbb{R}^{n \\times n},\\ \\lambda_1, \\ldots, \\lambda_n \\in \\mathbb{C} \\text{ alle Eigenwerte mit Vielfachheit}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmsup>\u003Cmi mathvariant=\"double-struck\">R\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>×\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsup>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A \\in \\mathbb{R}^{n \\times n},\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7224em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9658em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathbb\">R\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7713em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">×\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">C\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lambda_1, \\ldots, \\lambda_n \\in \\mathbb{C}\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6889em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathbb\">C\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">alle Eigenwerte mit Vielfachheit\u003C\u002Fspan>",null,"md","eigenvalue-properties","\\operatorname{Spur}(A) = \\lambda_1 + \\lambda_2 + \\ldots + \\lambda_n, \\qquad \\det A = \\lambda_1 \\cdot \\lambda_2 \\cdots \\lambda_n","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Spur\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"2em\"\u002F>\u003Cmi>det\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>⋯\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{Spur}(A) = \\lambda_1 + \\lambda_2 + \\ldots + \\lambda_n, \\qquad \\det A = \\lambda_1 \\cdot \\lambda_2 \\cdots \\lambda_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">Spur\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:2em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">det\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8444em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">⋯\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspur-und-determinante-eigenwerte",[],[4052,4053,4054,4055],"characteristic-polynomial","matrix-trace","determinant-rules","diagonalization",{"title":548,"description":4005},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspur-und-determinante-eigenwerte","Die Summe aller Eigenwerte einer Matrix ist ihre Spur, das Produkt aller Eigenwerte ihre Determinante.",[4060,4061,4062,4063],"eigenwert","spur","determinante","charakteristisches polynom",[4065,4068,4072,4076],{"symbol":706,"meaning":4066,"symbolHtml":4067},"quadratische n × n-Matrix","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4069,"meaning":4070,"symbolHtml":4071},"\\lambda_1, \\ldots, \\lambda_n","Eigenwerte von A, nach algebraischer Vielfachheit wiederholt, eventuell komplex","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>λ\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\lambda_1, \\ldots, \\lambda_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"minner\">…\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4073,"meaning":4074,"symbolHtml":4075},"\\operatorname{Spur}(A)","Summe der Diagonaleinträge von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi mathvariant=\"normal\">Spur\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{Spur}(A)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\">Spur\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":1143,"meaning":4077,"symbolHtml":4078},"Determinante von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>det\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>A\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\det A\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">det\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","UBZmtVMk5V8o8d_wozzKd5-hyQkmaCLVz8S8TA26mUk",[4081,4084,4087,4090],{"stem":4082,"title":4083,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fdiagonalisierung","Diagonalisierung",{"stem":4085,"title":4086,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Feigenwerte-charakteristisches-polynom","Eigenwerte und charakteristisches Polynom",{"stem":4088,"title":4089,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Frechenregeln-determinanten","Rechenregeln für Determinanten",{"stem":4091,"title":4092,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspur-einer-matrix","Spur einer Matrix",1793824280220]