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\\operatorname{Spur}(BA)",[586,1071,1073,1106],{"className":1072,"ariaHidden":662},[661],[586,1074,1076,1079,1085,1088,1091,1094,1097,1100,1103],{"className":1075},[666],[586,1077],{"className":1078,"style":671},[670],[586,1080,1082],{"className":1081},[675],[586,1083,570],{"className":1084},[679,680],[586,1086,615],{"className":1087},[684],[586,1089,618],{"className":1090},[679,688],[586,1092,624],{"className":1093,"style":710},[679,688],[586,1095,627],{"className":1096},[714],[586,1098],{"className":1099,"style":718},[692],[586,1101,630],{"className":1102},[722],[586,1104],{"className":1105,"style":718},[692],[586,1107,1109,1112,1118,1121,1124,1127],{"className":1108},[666],[586,1110],{"className":1111,"style":671},[670],[586,1113,1115],{"className":1114},[675],[586,1116,570],{"className":1117},[679,680],[586,1119,615],{"className":1120},[684],[586,1122,624],{"className":1123,"style":710},[679,688],[586,1125,618],{"className":1126},[679,688],[586,1128,627],{"className":1129},[714],", sogar wenn ",[586,1132,1134,1168],{"className":1133},[589],[586,1135,1137],{"className":1136},[593],[595,1138,1139],{"xmlns":597},[599,1140,1141,1165],{},[602,1142,1143,1145,1148],{},[605,1144,618],{},[609,1146,1147],{},"∈",[906,1149,1150,1154],{},[605,1151,1153],{"mathvariant":1152},"double-struck","R",[602,1155,1156,1159,1162],{},[605,1157,1158],{},"m",[609,1160,1161],{},"×",[605,1163,1164],{},"n",[654,1166,1167],{"encoding":656},"A \\in \\mathbb{R}^{m \\times n}",[586,1169,1171,1190],{"className":1170,"ariaHidden":662},[661],[586,1172,1174,1178,1181,1184,1187],{"className":1173},[666],[586,1175],{"className":1176,"style":1177},[670],"height:0.7224em;vertical-align:-0.0391em;",[586,1179,618],{"className":1180},[679,688],[586,1182],{"className":1183,"style":718},[692],[586,1185,1147],{"className":1186},[722],[586,1188],{"className":1189,"style":718},[692],[586,1191,1193,1197],{"className":1192},[666],[586,1194],{"className":1195,"style":1196},[670],"height:0.7713em;",[586,1198,1200,1204],{"className":1199},[679],[586,1201,1153],{"className":1202},[679,1203],"mathbb",[586,1205,1207],{"className":1206},[958],[586,1208,1210],{"className":1209},[962],[586,1211,1213],{"className":1212},[966],[586,1214,1216],{"className":1215,"style":1196},[970],[586,1217,1218,1221],{"style":974},[586,1219],{"className":1220,"style":979},[978],[586,1222,1224],{"className":1223},[983,984,985,986],[586,1225,1227,1230,1233],{"className":1226},[679,986],[586,1228,1158],{"className":1229},[679,688,986],[586,1231,1161],{"className":1232},[697,986],[586,1234,1164],{"className":1235},[679,688,986],[586,1237,1239,1267],{"className":1238},[589],[586,1240,1242],{"className":1241},[593],[595,1243,1244],{"xmlns":597},[599,1245,1246,1264],{},[602,1247,1248,1250,1252],{},[605,1249,624],{},[609,1251,1147],{},[906,1253,1254,1256],{},[605,1255,1153],{"mathvariant":1152},[602,1257,1258,1260,1262],{},[605,1259,1164],{},[609,1261,1161],{},[605,1263,1158],{},[654,1265,1266],{"encoding":656},"B \\in \\mathbb{R}^{n \\times m}",[586,1268,1270,1288],{"className":1269,"ariaHidden":662},[661],[586,1271,1273,1276,1279,1282,1285],{"className":1272},[666],[586,1274],{"className":1275,"style":1177},[670],[586,1277,624],{"className":1278,"style":710},[679,688],[586,1280],{"className":1281,"style":718},[692],[586,1283,1147],{"className":1284},[722],[586,1286],{"className":1287,"style":718},[692],[586,1289,1291,1294],{"className":1290},[666],[586,1292],{"className":1293,"style":1196},[670],[586,1295,1297,1300],{"className":1296},[679],[586,1298,1153],{"className":1299},[679,1203],[586,1301,1303],{"className":1302},[958],[586,1304,1306],{"className":1305},[962],[586,1307,1309],{"className":1308},[966],[586,1310,1312],{"className":1311,"style":1196},[970],[586,1313,1314,1317],{"style":974},[586,1315],{"className":1316,"style":979},[978],[586,1318,1320],{"className":1319},[983,984,985,986],[586,1321,1323,1326,1329],{"className":1322},[679,986],[586,1324,1164],{"className":1325},[679,688,986],[586,1327,1161],{"className":1328},[697,986],[586,1330,1158],{"className":1331},[679,688,986]," verschiedene Größen haben.",[579,1334,1335,584,1338,1486,1487,1516],{},[568,1336,1337],{},"Ähnlichkeitsinvarianz:",[586,1339,1341,1391],{"className":1340},[589],[586,1342,1344],{"className":1343},[593],[595,1345,1346],{"xmlns":597},[599,1347,1348,1388],{},[602,1349,1350,1352,1354,1356,1370,1372,1374,1376,1378,1380,1382,1384,1386],{},[605,1351,570],{"mathvariant":607},[609,1353,611],{},[609,1355,615],{"stretchy":614},[906,1357,1358,1361],{},[605,1359,1360],{},"S",[602,1362,1363,1366],{},[609,1364,1365],{},"−",[1367,1368,1369],"mn",{},"1",[605,1371,618],{},[605,1373,1360],{},[609,1375,627],{"stretchy":614},[609,1377,630],{},[605,1379,570],{"mathvariant":607},[609,1381,611],{},[609,1383,615],{"stretchy":614},[605,1385,618],{},[609,1387,627],{"stretchy":614},[654,1389,1390],{"encoding":656},"\\operatorname{Spur}(S^{-1} A S) = \\operatorname{Spur}(A)",[586,1392,1394,1465],{"className":1393,"ariaHidden":662},[661],[586,1395,1397,1401,1407,1410,1447,1450,1453,1456,1459,1462],{"className":1396},[666],[586,1398],{"className":1399,"style":1400},[670],"height:1.0641em;vertical-align:-0.25em;",[586,1402,1404],{"className":1403},[675],[586,1405,570],{"className":1406},[679,680],[586,1408,615],{"className":1409},[684],[586,1411,1413,1417],{"className":1412},[679],[586,1414,1360],{"className":1415,"style":1416},[679,688],"margin-right:0.0576em;",[586,1418,1420],{"className":1419},[958],[586,1421,1423],{"className":1422},[962],[586,1424,1426],{"className":1425},[966],[586,1427,1430],{"className":1428,"style":1429},[970],"height:0.8141em;",[586,1431,1432,1435],{"style":974},[586,1433],{"className":1434,"style":979},[978],[586,1436,1438],{"className":1437},[983,984,985,986],[586,1439,1441,1444],{"className":1440},[679,986],[586,1442,1365],{"className":1443},[679,986],[586,1445,1369],{"className":1446},[679,986],[586,1448,618],{"className":1449},[679,688],[586,1451,1360],{"className":1452,"style":1416},[679,688],[586,1454,627],{"className":1455},[714],[586,1457],{"className":1458,"style":718},[692],[586,1460,630],{"className":1461},[722],[586,1463],{"className":1464,"style":718},[692],[586,1466,1468,1471,1477,1480,1483],{"className":1467},[666],[586,1469],{"className":1470,"style":671},[670],[586,1472,1474],{"className":1473},[675],[586,1475,570],{"className":1476},[679,680],[586,1478,615],{"className":1479},[684],[586,1481,618],{"className":1482},[679,688],[586,1484,627],{"className":1485},[714]," für invertierbares ",[586,1488,1490,1503],{"className":1489},[589],[586,1491,1493],{"className":1492},[593],[595,1494,1495],{"xmlns":597},[599,1496,1497,1501],{},[602,1498,1499],{},[605,1500,1360],{},[654,1502,1360],{"encoding":656},[586,1504,1506],{"className":1505,"ariaHidden":662},[661],[586,1507,1509,1513],{"className":1508},[666],[586,1510],{"className":1511,"style":1512},[670],"height:0.6833em;",[586,1514,1360],{"className":1515,"style":1416},[679,688],". Die Spur hängt also nur von der linearen Abbildung ab, nicht von der Basis.",[579,1518,1519,1520,1523,1524,1555],{},"Die Spur ist die ",[568,1521,1522],{},"Summe der Eigenwerte",", mit Vielfachheit und über ",[586,1525,1527,1542],{"className":1526},[589],[586,1528,1530],{"className":1529},[593],[595,1531,1532],{"xmlns":597},[599,1533,1534,1539],{},[602,1535,1536],{},[605,1537,1538],{"mathvariant":1152},"C",[654,1540,1541],{"encoding":656},"\\mathbb{C}",[586,1543,1545],{"className":1544,"ariaHidden":662},[661],[586,1546,1548,1552],{"className":1547},[666],[586,1549],{"className":1550,"style":1551},[670],"height:0.6889em;",[586,1553,1538],{"className":1554},[679,1203]," gezählt.",[564,1557,1558,1559,1562,1563,1727,1728,884],{},"Achtung: Bei drei oder mehr Faktoren darf man nur ",[568,1560,1561],{},"zyklisch"," vertauschen: ",[586,1564,1566,1624],{"className":1565},[589],[586,1567,1569],{"className":1568},[593],[595,1570,1571],{"xmlns":597},[599,1572,1573,1621],{},[602,1574,1575,1577,1579,1581,1583,1585,1587,1589,1591,1593,1595,1597,1599,1601,1603,1605,1607,1609,1611,1613,1615,1617,1619],{},[605,1576,570],{"mathvariant":607},[609,1578,611],{},[609,1580,615],{"stretchy":614},[605,1582,618],{},[605,1584,624],{},[605,1586,1538],{},[609,1588,627],{"stretchy":614},[609,1590,630],{},[605,1592,570],{"mathvariant":607},[609,1594,611],{},[609,1596,615],{"stretchy":614},[605,1598,624],{},[605,1600,1538],{},[605,1602,618],{},[609,1604,627],{"stretchy":614},[609,1606,630],{},[605,1608,570],{"mathvariant":607},[609,1610,611],{},[609,1612,615],{"stretchy":614},[605,1614,1538],{},[605,1616,618],{},[605,1618,624],{},[609,1620,627],{"stretchy":614},[654,1622,1623],{"encoding":656},"\\operatorname{Spur}(ABC) = \\operatorname{Spur}(BCA) = \\operatorname{Spur}(CAB)",[586,1625,1627,1664,1700],{"className":1626,"ariaHidden":662},[661],[586,1628,1630,1633,1639,1642,1645,1648,1652,1655,1658,1661],{"className":1629},[666],[586,1631],{"className":1632,"style":671},[670],[586,1634,1636],{"className":1635},[675],[586,1637,570],{"className":1638},[679,680],[586,1640,615],{"className":1641},[684],[586,1643,618],{"className":1644},[679,688],[586,1646,624],{"className":1647,"style":710},[679,688],[586,1649,1538],{"className":1650,"style":1651},[679,688],"margin-right:0.0715em;",[586,1653,627],{"className":1654},[714],[586,1656],{"className":1657,"style":718},[692],[586,1659,630],{"className":1660},[722],[586,1662],{"className":1663,"style":718},[692],[586,1665,1667,1670,1676,1679,1682,1685,1688,1691,1694,1697],{"className":1666},[666],[586,1668],{"className":1669,"style":671},[670],[586,1671,1673],{"className":1672},[675],[586,1674,570],{"className":1675},[679,680],[586,1677,615],{"className":1678},[684],[586,1680,624],{"className":1681,"style":710},[679,688],[586,1683,1538],{"className":1684,"style":1651},[679,688],[586,1686,618],{"className":1687},[679,688],[586,1689,627],{"className":1690},[714],[586,1692],{"className":1693,"style":718},[692],[586,1695,630],{"className":1696},[722],[586,1698],{"className":1699,"style":718},[692],[586,1701,1703,1706,1712,1715,1718,1721,1724],{"className":1702},[666],[586,1704],{"className":1705,"style":671},[670],[586,1707,1709],{"className":1708},[675],[586,1710,570],{"className":1711},[679,680],[586,1713,615],{"className":1714},[684],[586,1716,1538],{"className":1717,"style":1651},[679,688],[586,1719,618],{"className":1720},[679,688],[586,1722,624],{"className":1723,"style":710},[679,688],[586,1725,627],{"className":1726},[714],", aber im Allgemeinen nicht gleich ",[586,1729,1731,1757],{"className":1730},[589],[586,1732,1734],{"className":1733},[593],[595,1735,1736],{"xmlns":597},[599,1737,1738,1754],{},[602,1739,1740,1742,1744,1746,1748,1750,1752],{},[605,1741,570],{"mathvariant":607},[609,1743,611],{},[609,1745,615],{"stretchy":614},[605,1747,618],{},[605,1749,1538],{},[605,1751,624],{},[609,1753,627],{"stretchy":614},[654,1755,1756],{"encoding":656},"\\operatorname{Spur}(ACB)",[586,1758,1760],{"className":1759,"ariaHidden":662},[661],[586,1761,1763,1766,1772,1775,1778,1781,1784],{"className":1762},[666],[586,1764],{"className":1765,"style":671},[670],[586,1767,1769],{"className":1768},[675],[586,1770,570],{"className":1771},[679,680],[586,1773,615],{"className":1774},[684],[586,1776,618],{"className":1777},[679,688],[586,1779,1538],{"className":1780,"style":1651},[679,688],[586,1782,624],{"className":1783,"style":710},[679,688],[586,1785,627],{"className":1786},[714],[559,1788,1790],{"id":1789},"beispiele","Beispiele",[586,1792,1795],{"className":1793},[1794],"katex-display",[586,1796,1798,1922],{"className":1797},[589],[586,1799,1801],{"className":1800},[593],[595,1802,1804],{"xmlns":597,"display":1803},"block",[599,1805,1806,1919],{},[602,1807,1808,1810,1812,1897,1899,1901,1903,1905,1907,1909,1911,1913,1915,1917],{},[605,1809,570],{"mathvariant":607},[609,1811,611],{},[602,1813,1814,1816,1895],{},[609,1815,615],{"fence":662},[1817,1818,1822,1847,1873],"mtable",{"rowspacing":1819,"columnalign":1820,"columnspacing":1821},"0.16em","center center center","1em",[1823,1824,1825,1835,1841],"mtr",{},[1826,1827,1828],"mtd",{},[1829,1830,1832],"mstyle",{"scriptlevel":1831,"displaystyle":614},"0",[1367,1833,1834],{},"4",[1826,1836,1837],{},[1829,1838,1839],{"scriptlevel":1831,"displaystyle":614},[1367,1840,1369],{},[1826,1842,1843],{},[1829,1844,1845],{"scriptlevel":1831,"displaystyle":614},[1367,1846,1831],{},[1823,1848,1849,1856,1866],{},[1826,1850,1851],{},[1829,1852,1853],{"scriptlevel":1831,"displaystyle":614},[1367,1854,1855],{},"2",[1826,1857,1858],{},[1829,1859,1860],{"scriptlevel":1831,"displaystyle":614},[602,1861,1862,1864],{},[609,1863,1365],{},[1367,1865,1369],{},[1826,1867,1868],{},[1829,1869,1870],{"scriptlevel":1831,"displaystyle":614},[1367,1871,1872],{},"5",[1823,1874,1875,1882,1889],{},[1826,1876,1877],{},[1829,1878,1879],{"scriptlevel":1831,"displaystyle":614},[1367,1880,1881],{},"3",[1826,1883,1884],{},[1829,1885,1886],{"scriptlevel":1831,"displaystyle":614},[1367,1887,1888],{},"7",[1826,1890,1891],{},[1829,1892,1893],{"scriptlevel":1831,"displaystyle":614},[1367,1894,1855],{},[609,1896,627],{"fence":662},[609,1898,630],{},[1367,1900,1834],{},[609,1902,621],{},[609,1904,615],{"stretchy":614},[609,1906,1365],{},[1367,1908,1369],{},[609,1910,627],{"stretchy":614},[609,1912,621],{},[1367,1914,1855],{},[609,1916,630],{},[1367,1918,1872],{},[654,1920,1921],{"encoding":656},"\\operatorname{Spur} \\begin{pmatrix} 4 & 1 & 0 \\\\ 2 & -1 & 5 \\\\ 3 & 7 & 2 \\end{pmatrix} = 4 + (-1) + 2 = 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ist ",[586,2888,2890,2945],{"className":2889},[589],[586,2891,2893],{"className":2892},[593],[595,2894,2895],{"xmlns":597},[599,2896,2897,2942],{},[602,2898,2899,2901,2903,2905],{},[605,2900,618],{},[605,2902,624],{},[609,2904,630],{},[602,2906,2907,2909,2940],{},[609,2908,615],{"fence":662},[1817,2910,2911,2925],{"rowspacing":1819,"columnalign":2610,"columnspacing":1821},[1823,2912,2913,2919],{},[1826,2914,2915],{},[1829,2916,2917],{"scriptlevel":1831,"displaystyle":614},[1367,2918,1872],{},[1826,2920,2921],{},[1829,2922,2923],{"scriptlevel":1831,"displaystyle":614},[1367,2924,1855],{},[1823,2926,2927,2933],{},[1826,2928,2929],{},[1829,2930,2931],{"scriptlevel":1831,"displaystyle":614},[1367,2932,1872],{},[1826,2934,2935],{},[1829,2936,2937],{"scriptlevel":1831,"displaystyle":614},[1367,2938,2939],{},"13",[609,2941,627],{"fence":662},[654,2943,2944],{"encoding":656},"AB = \\begin{pmatrix} 5 & 2 \\\\ 5 & 13 \\end{pmatrix}",[586,2946,2948,2969],{"className":2947,"ariaHidden":662},[661],[586,2949,2951,2954,2957,2960,2963,2966],{"className":2950},[666],[586,2952],{"className":2953,"style":1512},[670],[586,2955,618],{"className":2956},[679,688],[586,2958,624],{"className":2959,"style":710},[679,688],[586,2961],{"className":2962,"style":718},[692],[586,2964,630],{"className":2965},[722],[586,2967],{"className":2968,"style":718},[692],[586,2970,2972,2975],{"className":2971},[666],[586,2973],{"className":2974,"style":2413},[670],[586,2976,2978,2984,3086],{"className":2977},[1944],[586,2979,2981],{"className":2980,"style":2421},[684,2420],[586,2982,615],{"className":2983},[1951,985],[586,2985,2987],{"className":2986},[679],[586,2988,2990,3035,3038,3041],{"className":2989},[1817],[586,2991,2993],{"className":2992},[2008],[586,2994,2996,3027],{"className":2995},[962,1956],[586,2997,2999,3024],{"className":2998},[966],[586,3000,3002,3013],{"className":3001,"style":2443},[970],[586,3003,3004,3007],{"style":2446},[586,3005],{"className":3006,"style":2024},[978],[586,3008,3010],{"className":3009},[679],[586,3011,1872],{"className":3012},[679],[586,3014,3015,3018],{"style":2458},[586,3016],{"className":3017,"style":2024},[978],[586,3019,3021],{"className":3020},[679],[586,3022,1872],{"className":3023},[679],[586,3025,1989],{"className":3026},[1988],[586,3028,3030],{"className":3029},[966],[586,3031,3033],{"className":3032,"style":2477},[970],[586,3034],{},[586,3036],{"className":3037,"style":2070},[2069],[586,3039],{"className":3040,"style":2070},[2069],[586,3042,3044],{"className":3043},[2008],[586,3045,3047,3078],{"className":3046},[962,1956],[586,3048,3050,3075],{"className":3049},[966],[586,3051,3053,3064],{"className":3052,"style":2443},[970],[586,3054,3055,3058],{"style":2446},[586,3056],{"className":3057,"style":2024},[978],[586,3059,3061],{"className":3060},[679],[586,3062,1855],{"className":3063},[679],[586,3065,3066,3069],{"style":2458},[586,3067],{"className":3068,"style":2024},[978],[586,3070,3072],{"className":3071},[679],[586,3073,2939],{"className":3074},[679],[586,3076,1989],{"className":3077},[1988],[586,3079,3081],{"className":3080},[966],[586,3082,3084],{"className":3083,"style":2477},[970],[586,3085],{},[586,3087,3089],{"className":3088,"style":2421},[714,2420],[586,3090,627],{"className":3091},[1951,985]," eine ",[586,3094,3096,3114],{"className":3095},[589],[586,3097,3099],{"className":3098},[593],[595,3100,3101],{"xmlns":597},[599,3102,3103,3111],{},[602,3104,3105,3107,3109],{},[1367,3106,1855],{},[609,3108,1161],{},[1367,3110,1855],{},[654,3112,3113],{"encoding":656},"2 \\times 2",[586,3115,3117,3135],{"className":3116,"ariaHidden":662},[661],[586,3118,3120,3123,3126,3129,3132],{"className":3119},[666],[586,3121],{"className":3122,"style":2248},[670],[586,3124,1855],{"className":3125},[679],[586,3127],{"className":3128,"style":693},[692],[586,3130,1161],{"className":3131},[697],[586,3133],{"className":3134,"style":693},[692],[586,3136,3138,3141],{"className":3137},[666],[586,3139],{"className":3140,"style":2294},[670],[586,3142,1855],{"className":3143},[679],"-Matrix und ",[586,3146,3148,3236],{"className":3147},[589],[586,3149,3151],{"className":3150},[593],[595,3152,3153],{"xmlns":597},[599,3154,3155,3233],{},[602,3156,3157,3159,3161,3163],{},[605,3158,624],{},[605,3160,618],{},[609,3162,630],{},[602,3164,3165,3167,3231],{},[609,3166,615],{"fence":662},[1817,3168,3169,3189,3209],{"rowspacing":1819,"columnalign":1820,"columnspacing":1821},[1823,3170,3171,3177,3183],{},[1826,3172,3173],{},[1829,3174,3175],{"scriptlevel":1831,"displaystyle":614},[1367,3176,1369],{},[1826,3178,3179],{},[1829,3180,3181],{"scriptlevel":1831,"displaystyle":614},[1367,3182,1855],{},[1826,3184,3185],{},[1829,3186,3187],{"scriptlevel":1831,"displaystyle":614},[1367,3188,1831],{},[1823,3190,3191,3197,3203],{},[1826,3192,3193],{},[1829,3194,3195],{"scriptlevel":1831,"displaystyle":614},[1367,3196,1872],{},[1826,3198,3199],{},[1829,3200,3201],{"scriptlevel":1831,"displaystyle":614},[1367,3202,1872],{},[1826,3204,3205],{},[1829,3206,3207],{"scriptlevel":1831,"displaystyle":614},[1367,3208,1834],{},[1823,3210,3211,3218,3224],{},[1826,3212,3213],{},[1829,3214,3215],{"scriptlevel":1831,"displaystyle":614},[1367,3216,3217],{},"9",[1826,3219,3220],{},[1829,3221,3222],{"scriptlevel":1831,"displaystyle":614},[1367,3223,1881],{},[1826,3225,3226],{},[1829,3227,3228],{"scriptlevel":1831,"displaystyle":614},[1367,3229,3230],{},"12",[609,3232,627],{"fence":662},[654,3234,3235],{"encoding":656},"BA = \\begin{pmatrix} 1 & 2 & 0 \\\\ 5 & 5 & 4 \\\\ 9 & 3 & 12 \\end{pmatrix}",[586,3237,3239,3260],{"className":3238,"ariaHidden":662},[661],[586,3240,3242,3245,3248,3251,3254,3257],{"className":3241},[666],[586,3243],{"className":3244,"style":1512},[670],[586,3246,624],{"className":3247,"style":710},[679,688],[586,3249,618],{"className":3250},[679,688],[586,3252],{"className":3253,"style":718},[692],[586,3255,630],{"className":3256},[722],[586,3258],{"className":3259,"style":718},[692],[586,3261,3263,3266],{"className":3262},[666],[586,3264],{"className":3265,"style":1931},[670],[586,3267,3269,3306,3492],{"className":3268},[1944],[586,3270,3272],{"className":3271},[684],[586,3273,3275],{"className":3274},[1951,1952],[586,3276,3278,3298],{"className":3277},[962,1956],[586,3279,3281,3295],{"className":3280},[966],[586,3282,3284],{"className":3283,"style":1963},[970],[586,3285,3286,3289],{"style":1966},[586,3287],{"className":3288,"style":1970},[978],[586,3290,3291],{"style":1973},[1975,3292,3293],{"xmlns":1977,"width":1978,"height":1979,"viewBox":1980},[1982,3294],{"d":1984},[586,3296,1989],{"className":3297},[1988],[586,3299,3301],{"className":3300},[966],[586,3302,3304],{"className":3303,"style":1996},[970],[586,3305],{},[586,3307,3309],{"className":3308},[679],[586,3310,3312,3368,3371,3374,3430,3433,3436],{"className":3311},[1817],[586,3313,3315],{"className":3314},[2008],[586,3316,3318,3360],{"className":3317},[962,1956],[586,3319,3321,3357],{"className":3320},[966],[586,3322,3324,3335,3346],{"className":3323,"style":1963},[970],[586,3325,3326,3329],{"style":2020},[586,3327],{"className":3328,"style":2024},[978],[586,3330,3332],{"className":3331},[679],[586,3333,1369],{"className":3334},[679],[586,3336,3337,3340],{"style":2033},[586,3338],{"className":3339,"style":2024},[978],[586,3341,3343],{"className":3342},[679],[586,3344,1872],{"className":3345},[679],[586,3347,3348,3351],{"style":2045},[586,3349],{"className":3350,"style":2024},[978],[586,3352,3354],{"className":3353},[679],[586,3355,3217],{"className":3356},[679],[586,3358,1989],{"className":3359},[1988],[586,3361,3363],{"className":3362},[966],[586,3364,3366],{"className":3365,"style":1996},[970],[586,3367],{},[586,3369],{"className":3370,"style":2070},[2069],[586,3372],{"className":3373,"style":2070},[2069],[586,3375,3377],{"className":3376},[2008],[586,3378,3380,3422],{"className":3379},[962,1956],[586,3381,3383,3419],{"className":3382},[966],[586,3384,3386,3397,3408],{"className":3385,"style":1963},[970],[586,3387,3388,3391],{"style":2020},[586,3389],{"className":3390,"style":2024},[978],[586,3392,3394],{"className":3393},[679],[586,3395,1855],{"className":3396},[679],[586,3398,3399,3402],{"style":2033},[586,3400],{"className":3401,"style":2024},[978],[586,3403,3405],{"className":3404},[679],[586,3406,1872],{"className":3407},[679],[586,3409,3410,3413],{"style":2045},[586,3411],{"className":3412,"style":2024},[978],[586,3414,3416],{"className":3415},[679],[586,3417,1881],{"className":3418},[679],[586,3420,1989],{"className":3421},[1988],[586,3423,3425],{"className":3424},[966],[586,3426,3428],{"className":3427,"style":1996},[970],[586,3429],{},[586,3431],{"className":3432,"style":2070},[2069],[586,3434],{"className":3435,"style":2070},[2069],[586,3437,3439],{"className":3438},[2008],[586,3440,3442,3484],{"className":3441},[962,1956],[586,3443,3445,3481],{"className":3444},[966],[586,3446,3448,3459,3470],{"className":3447,"style":1963},[970],[586,3449,3450,3453],{"style":2020},[586,3451],{"className":3452,"style":2024},[978],[586,3454,3456],{"className":3455},[679],[586,3457,1831],{"className":3458},[679],[586,3460,3461,3464],{"style":2033},[586,3462],{"className":3463,"style":2024},[978],[586,3465,3467],{"className":3466},[679],[586,3468,1834],{"className":3469},[679],[586,3471,3472,3475],{"style":2045},[586,3473],{"className":3474,"style":2024},[978],[586,3476,3478],{"className":3477},[679],[586,3479,3230],{"className":3480},[679],[586,3482,1989],{"className":3483},[1988],[586,3485,3487],{"className":3486},[966],[586,3488,3490],{"className":3489,"style":1996},[970],[586,3491],{},[586,3493,3495],{"className":3494},[714],[586,3496,3498],{"className":3497},[1951,1952],[586,3499,3501,3521],{"className":3500},[962,1956],[586,3502,3504,3518],{"className":3503},[966],[586,3505,3507],{"className":3506,"style":1963},[970],[586,3508,3509,3512],{"style":1966},[586,3510],{"className":3511,"style":1970},[978],[586,3513,3514],{"style":1973},[1975,3515,3516],{"xmlns":1977,"width":1978,"height":1979,"viewBox":1980},[1982,3517],{"d":2221},[586,3519,1989],{"className":3520},[1988],[586,3522,3524],{"className":3523},[966],[586,3525,3527],{"className":3526,"style":1996},[970],[586,3528],{},[586,3530,3532,3550],{"className":3531},[589],[586,3533,3535],{"className":3534},[593],[595,3536,3537],{"xmlns":597},[599,3538,3539,3547],{},[602,3540,3541,3543,3545],{},[1367,3542,1881],{},[609,3544,1161],{},[1367,3546,1881],{},[654,3548,3549],{"encoding":656},"3 \\times 3",[586,3551,3553,3571],{"className":3552,"ariaHidden":662},[661],[586,3554,3556,3559,3562,3565,3568],{"className":3555},[666],[586,3557],{"className":3558,"style":2248},[670],[586,3560,1881],{"className":3561},[679],[586,3563],{"className":3564,"style":693},[692],[586,3566,1161],{"className":3567},[697],[586,3569],{"className":3570,"style":693},[692],[586,3572,3574,3577],{"className":3573},[666],[586,3575],{"className":3576,"style":2294},[670],[586,3578,1881],{"className":3579},[679],"-Matrix. Trotzdem stimmen die Spuren überein: ",[586,3582,3584,3619],{"className":3583},[589],[586,3585,3587],{"className":3586},[593],[595,3588,3589],{"xmlns":597},[599,3590,3591,3616],{},[602,3592,3593,3595,3597,3599,3601,3604,3606,3608,3610,3612,3614],{},[1367,3594,1872],{},[609,3596,621],{},[1367,3598,2939],{},[609,3600,630],{},[1367,3602,3603],{},"18",[609,3605,630],{},[1367,3607,1369],{},[609,3609,621],{},[1367,3611,1872],{},[609,3613,621],{},[1367,3615,3230],{},[654,3617,3618],{"encoding":656},"5 + 13 = 18 = 1 + 5 + 12",[586,3620,3622,3640,3658,3676,3694,3712],{"className":3621,"ariaHidden":662},[661],[586,3623,3625,3628,3631,3634,3637],{"className":3624},[666],[586,3626],{"className":3627,"style":2248},[670],[586,3629,1872],{"className":3630},[679],[586,3632],{"className":3633,"style":693},[692],[586,3635,621],{"className":3636},[697],[586,3638],{"className":3639,"style":693},[692],[586,3641,3643,3646,3649,3652,3655],{"className":3642},[666],[586,3644],{"className":3645,"style":2294},[670],[586,3647,2939],{"className":3648},[679],[586,3650],{"className":3651,"style":718},[692],[586,3653,630],{"className":3654},[722],[586,3656],{"className":3657,"style":718},[692],[586,3659,3661,3664,3667,3670,3673],{"className":3660},[666],[586,3662],{"className":3663,"style":2294},[670],[586,3665,3603],{"className":3666},[679],[586,3668],{"className":3669,"style":718},[692],[586,3671,630],{"className":3672},[722],[586,3674],{"className":3675,"style":718},[692],[586,3677,3679,3682,3685,3688,3691],{"className":3678},[666],[586,3680],{"className":3681,"style":2248},[670],[586,3683,1369],{"className":3684},[679],[586,3686],{"className":3687,"style":693},[692],[586,3689,621],{"className":3690},[697],[586,3692],{"className":3693,"style":693},[692],[586,3695,3697,3700,3703,3706,3709],{"className":3696},[666],[586,3698],{"className":3699,"style":2248},[670],[586,3701,1872],{"className":3702},[679],[586,3704],{"className":3705,"style":693},[692],[586,3707,621],{"className":3708},[697],[586,3710],{"className":3711,"style":693},[692],[586,3713,3715,3718],{"className":3714},[666],[586,3716],{"className":3717,"style":2294},[670],[586,3719,3230],{"className":3720},[679],[564,3722,3723,3724,3899,3900,777,3928,3956,3957,884],{},"Die Matrix ",[586,3725,3727,3773],{"className":3726},[589],[586,3728,3730],{"className":3729},[593],[595,3731,3732],{"xmlns":597},[599,3733,3734,3770],{},[602,3735,3736,3738,3768],{},[609,3737,615],{"fence":662},[1817,3739,3740,3754],{"rowspacing":1819,"columnalign":2610,"columnspacing":1821},[1823,3741,3742,3748],{},[1826,3743,3744],{},[1829,3745,3746],{"scriptlevel":1831,"displaystyle":614},[1367,3747,1834],{},[1826,3749,3750],{},[1829,3751,3752],{"scriptlevel":1831,"displaystyle":614},[1367,3753,1369],{},[1823,3755,3756,3762],{},[1826,3757,3758],{},[1829,3759,3760],{"scriptlevel":1831,"displaystyle":614},[1367,3761,1855],{},[1826,3763,3764],{},[1829,3765,3766],{"scriptlevel":1831,"displaystyle":614},[1367,3767,1881],{},[609,3769,627],{"fence":662},[654,3771,3772],{"encoding":656},"\\begin{pmatrix} 4 & 1 \\\\ 2 & 3 \\end{pmatrix}",[586,3774,3776],{"className":3775,"ariaHidden":662},[661],[586,3777,3779,3782],{"className":3778},[666],[586,3780],{"className":3781,"style":2413},[670],[586,3783,3785,3791,3893],{"className":3784},[1944],[586,3786,3788],{"className":3787,"style":2421},[684,2420],[586,3789,615],{"className":3790},[1951,985],[586,3792,3794],{"className":3793},[679],[586,3795,3797,3842,3845,3848],{"className":3796},[1817],[586,3798,3800],{"className":3799},[2008],[586,3801,3803,3834],{"className":3802},[962,1956],[586,3804,3806,3831],{"className":3805},[966],[586,3807,3809,3820],{"className":3808,"style":2443},[970],[586,3810,3811,3814],{"style":2446},[586,3812],{"className":3813,"style":2024},[978],[586,3815,3817],{"className":3816},[679],[586,3818,1834],{"className":3819},[679],[586,3821,3822,3825],{"style":2458},[586,3823],{"className":3824,"style":2024},[978],[586,3826,3828],{"className":3827},[679],[586,3829,1855],{"className":3830},[679],[586,3832,1989],{"className":3833},[1988],[586,3835,3837],{"className":3836},[966],[586,3838,3840],{"className":3839,"style":2477},[970],[586,3841],{},[586,3843],{"className":3844,"style":2070},[2069],[586,3846],{"className":3847,"style":2070},[2069],[586,3849,3851],{"className":3850},[2008],[586,3852,3854,3885],{"className":3853},[962,1956],[586,3855,3857,3882],{"className":3856},[966],[586,3858,3860,3871],{"className":3859,"style":2443},[970],[586,3861,3862,3865],{"style":2446},[586,3863],{"className":3864,"style":2024},[978],[586,3866,3868],{"className":3867},[679],[586,3869,1369],{"className":3870},[679],[586,3872,3873,3876],{"style":2458},[586,3874],{"className":3875,"style":2024},[978],[586,3877,3879],{"className":3878},[679],[586,3880,1881],{"className":3881},[679],[586,3883,1989],{"className":3884},[1988],[586,3886,3888],{"className":3887},[966],[586,3889,3891],{"className":3890,"style":2477},[970],[586,3892],{},[586,3894,3896],{"className":3895,"style":2421},[714,2420],[586,3897,627],{"className":3898},[1951,985]," hat die Eigenwerte ",[586,3901,3903,3916],{"className":3902},[589],[586,3904,3906],{"className":3905},[593],[595,3907,3908],{"xmlns":597},[599,3909,3910,3914],{},[602,3911,3912],{},[1367,3913,1855],{},[654,3915,1855],{"encoding":656},[586,3917,3919],{"className":3918,"ariaHidden":662},[661],[586,3920,3922,3925],{"className":3921},[666],[586,3923],{"className":3924,"style":2294},[670],[586,3926,1855],{"className":3927},[679],[586,3929,3931,3944],{"className":3930},[589],[586,3932,3934],{"className":3933},[593],[595,3935,3936],{"xmlns":597},[599,3937,3938,3942],{},[602,3939,3940],{},[1367,3941,1872],{},[654,3943,1872],{"encoding":656},[586,3945,3947],{"className":3946,"ariaHidden":662},[661],[586,3948,3950,3953],{"className":3949},[666],[586,3951],{"className":3952,"style":2294},[670],[586,3954,1872],{"className":3955},[679],". Ihre Spur ist ",[586,3958,3960,3990],{"className":3959},[589],[586,3961,3963],{"className":3962},[593],[595,3964,3965],{"xmlns":597},[599,3966,3967,3987],{},[602,3968,3969,3971,3973,3975,3977,3979,3981,3983,3985],{},[1367,3970,1834],{},[609,3972,621],{},[1367,3974,1881],{},[609,3976,630],{},[1367,3978,1888],{},[609,3980,630],{},[1367,3982,1855],{},[609,3984,621],{},[1367,3986,1872],{},[654,3988,3989],{"encoding":656},"4 + 3 = 7 = 2 + 5",[586,3991,3993,4011,4029,4047,4065],{"className":3992,"ariaHidden":662},[661],[586,3994,3996,3999,4002,4005,4008],{"className":3995},[666],[586,3997],{"className":3998,"style":2248},[670],[586,4000,1834],{"className":4001},[679],[586,4003],{"className":4004,"style":693},[692],[586,4006,621],{"className":4007},[697],[586,4009],{"className":4010,"style":693},[692],[586,4012,4014,4017,4020,4023,4026],{"className":4013},[666],[586,4015],{"className":4016,"style":2294},[670],[586,4018,1881],{"className":4019},[679],[586,4021],{"className":4022,"style":718},[692],[586,4024,630],{"className":4025},[722],[586,4027],{"className":4028,"style":718},[692],[586,4030,4032,4035,4038,4041,4044],{"className":4031},[666],[586,4033],{"className":4034,"style":2294},[670],[586,4036,1888],{"className":4037},[679],[586,4039],{"className":4040,"style":718},[692],[586,4042,630],{"className":4043},[722],[586,4045],{"className":4046,"style":718},[692],[586,4048,4050,4053,4056,4059,4062],{"className":4049},[666],[586,4051],{"className":4052,"style":2248},[670],[586,4054,1855],{"className":4055},[679],[586,4057],{"className":4058,"style":693},[692],[586,4060,621],{"className":4061},[697],[586,4063],{"className":4064,"style":693},[692],[586,4066,4068,4071],{"className":4067},[666],[586,4069],{"className":4070,"style":2294},[670],[586,4072,1872],{"className":4073},[679],[559,4075,4077],{"id":4076},"typische-fehler","Typische Fehler",[576,4079,4080,4083,4225],{},[579,4081,4082],{},"Alle Einträge statt nur der Diagonaleinträge addieren.",[579,4084,4085,4224],{},[586,4086,4088,4137],{"className":4087},[589],[586,4089,4091],{"className":4090},[593],[595,4092,4093],{"xmlns":597},[599,4094,4095,4134],{},[602,4096,4097,4099,4101,4103,4105,4107,4109,4111,4113,4115,4117,4119,4121,4124,4126,4128,4130,4132],{},[605,4098,570],{"mathvariant":607},[609,4100,611],{},[609,4102,615],{"stretchy":614},[605,4104,618],{},[605,4106,624],{},[609,4108,627],{"stretchy":614},[609,4110,630],{},[605,4112,570],{"mathvariant":607},[609,4114,611],{},[609,4116,615],{"stretchy":614},[605,4118,618],{},[609,4120,627],{"stretchy":614},[609,4122,4123],{},"⋅",[605,4125,570],{"mathvariant":607},[609,4127,611],{},[609,4129,615],{"stretchy":614},[605,4131,624],{},[609,4133,627],{"stretchy":614},[654,4135,4136],{"encoding":656},"\\operatorname{Spur}(AB) = \\operatorname{Spur}(A) \\cdot \\operatorname{Spur}(B)",[586,4138,4140,4173,4203],{"className":4139,"ariaHidden":662},[661],[586,4141,4143,4146,4152,4155,4158,4161,4164,4167,4170],{"className":4142},[666],[586,4144],{"className":4145,"style":671},[670],[586,4147,4149],{"className":4148},[675],[586,4150,570],{"className":4151},[679,680],[586,4153,615],{"className":4154},[684],[586,4156,618],{"className":4157},[679,688],[586,4159,624],{"className":4160,"style":710},[679,688],[586,4162,627],{"className":4163},[714],[586,4165],{"className":4166,"style":718},[692],[586,4168,630],{"className":4169},[722],[586,4171],{"className":4172,"style":718},[692],[586,4174,4176,4179,4185,4188,4191,4194,4197,4200],{"className":4175},[666],[586,4177],{"className":4178,"style":671},[670],[586,4180,4182],{"className":4181},[675],[586,4183,570],{"className":4184},[679,680],[586,4186,615],{"className":4187},[684],[586,4189,618],{"className":4190},[679,688],[586,4192,627],{"className":4193},[714],[586,4195],{"className":4196,"style":693},[692],[586,4198,4123],{"className":4199},[697],[586,4201],{"className":4202,"style":693},[692],[586,4204,4206,4209,4215,4218,4221],{"className":4205},[666],[586,4207],{"className":4208,"style":671},[670],[586,4210,4212],{"className":4211},[675],[586,4213,570],{"className":4214},[679,680],[586,4216,615],{"className":4217},[684],[586,4219,624],{"className":4220,"style":710},[679,688],[586,4222,627],{"className":4223},[714]," annehmen. Die Spur ist additiv, aber nicht multiplikativ. Multiplikativ ist die Determinante.",[579,4226,4227],{},"Bei drei Faktoren beliebig vertauschen. Erlaubt sind nur zyklische Vertauschungen.",{"title":4229,"searchDepth":4230,"depth":4230,"links":4231},"",2,[4232,4233,4234],{"id":561,"depth":4230,"text":562},{"id":1789,"depth":4230,"text":1790},{"id":4076,"depth":4230,"text":4077},[4236,4239,4242,4245,4248,4250,4252,4254,4256,4258],{"lhs":4237,"rhs":4238},"trace([[a, b, c], [d, f, g], [h, k, l]])","a + f + l",{"lhs":4240,"rhs":4241},"trace([[a, b], [c, d]] + t*[[p, q], [r, s]])","trace([[a, b], [c, d]]) + t*trace([[p, q], [r, s]])",{"lhs":4243,"rhs":4244},"trace(multiply([[a, b, c], [d, f, g]], [[p, q], [r, s], [u, v]]))","trace(multiply([[p, q], [r, s], [u, v]], [[a, b, c], [d, f, g]]))",{"lhs":4246,"rhs":4247},"trace(multiply(inv([[2, 1], [1, 1]]), multiply([[a, b], [c, d]], [[2, 1], [1, 1]])))","a + d",{"lhs":4249,"rhs":1872},"trace([[4, 1, 0], [2, -1, 5], [3, 7, 2]])",{"lhs":4251,"rhs":3603},"trace(multiply([[1, 2, 0], [3, 1, 4]], [[1, 0], [2, 1], [0, 3]]))",{"lhs":4253,"rhs":3603},"5 + 13",{"lhs":4255,"rhs":3603},"trace(multiply([[1, 0], [2, 1], [0, 3]], [[1, 2, 0], [3, 1, 4]]))",{"lhs":4257,"rhs":3603},"1 + 5 + 12",{"lhs":4259,"rhs":4260},"4 + 3","2 + 5","A \\in \\mathbb{R}^{n \\times n}","\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>\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:0.7713em;\">\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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","matrix-trace","\\operatorname{Spur}(A) = \\sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \\ldots + a_{nn}","\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>\u003Cmunderover>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmunderover>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmn>11\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmn>22\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmo>…\u003C\u002Fmo>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\operatorname{Spur}(A) = \\sum_{i=1}^{n} a_{ii} = a_{11} + a_{22} + \\ldots + a_{nn}\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:2.9291em;vertical-align:-1.2777em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6514em;\">\u003Cspan style=\"top:-1.8723em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">i\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.05em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop op-symbol large-op\">∑\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-4.3em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3.05em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\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:1.2777em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\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\">\u003Cspan class=\"mord mathnormal mtight\">ii\u003C\u002Fspan>\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:0.7333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\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\">\u003Cspan class=\"mord mtight\">11\u003C\u002Fspan>\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:0.7333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\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\">\u003Cspan class=\"mord mtight\">22\u003C\u002Fspan>\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:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\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 mtight\">\u003Cspan class=\"mord mathnormal mtight\">nn\u003C\u002Fspan>\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-einer-matrix",[],[4274,4275,4276],"eigenvalue-properties","characteristic-polynomial","change-of-basis",{"title":548,"description":4229},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspur-einer-matrix","Die Spur einer quadratischen Matrix ist die Summe ihrer Diagonaleinträge; sie ist linear und es gilt Spur(AB) = Spur(BA).",[4281,4282,4283,4284],"spur","matrix","diagonale","eigenwert",[4286,4289,4293],{"symbol":618,"meaning":4287,"symbolHtml":4288},"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":4290,"meaning":4291,"symbolHtml":4292},"a_{ii}","Diagonaleinträge von A","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>i\u003C\u002Fmi>\u003Cmi>i\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a_{ii}\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:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3117em;\">\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\">\u003Cspan class=\"mord mathnormal mtight\">ii\u003C\u002Fspan>\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":4294,"meaning":4295,"symbolHtml":4296},"\\operatorname{Spur}(A)","Spur von A, auch tr(A) geschrieben","\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>","FM77T_Ewfvy_JbyfAqY_tGcX-cE1j0oz_VbwqES5fAM",[4299,4302,4305],{"stem":4300,"title":4301,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fbasiswechsel","Basiswechsel",{"stem":4303,"title":4304,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Feigenwerte-charakteristisches-polynom","Eigenwerte und charakteristisches Polynom",{"stem":4306,"title":4307,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspur-und-determinante-eigenwerte","Spur und Determinante als Summe und Produkt der Eigenwerte",1793824280154]