设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是斜对称矩阵当且仅当对于[tex=1.429x1.0]id8CqLD3sKgZOEL0mYn1xA==[/tex]中任一列向量[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex]，有[tex=3.571x1.143]Prw0L7uJ/bbBm5GTYZ6HIfXIkhPKaEJaPpuLa3Pkb6U=[/tex]。

2022-06-29

设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是斜对称矩阵当且仅当对于[tex=1.429x1.0]id8CqLD3sKgZOEL0mYn1xA==[/tex]中任一列向量[tex=0.643x0.786]SPoVA3bJlgfP9Ek9O4AbuA==[/tex]，有[tex=3.571x1.143]Prw0L7uJ/bbBm5GTYZ6HIfXIkhPKaEJaPpuLa3Pkb6U=[/tex]。

答案：

证明：必要性。设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是斜对称矩阵，则[tex=3.857x1.286]+4mjAfMHdXcM7vsa4fbsJg==[/tex]。于是[tex=11.571x1.5]0idGSV3RW/tbV3escumNdCmdGMiK3qcee3NUzUm5bkJ7/r4WLsCHgfnvhOgwwVKwdF9c98TPacwtQw+fMUwB9SgxCqeBSQFS6MR4wVLqRGNQZaLDKxZcviQC0y4KG6ZIB++zIoVswGbeyJt9vXjcvP8cBbM/Ou3l8Cz077EQ5WvQZ8NN8tSEEzlHtTG3U/yhmBlOAdrjbwBi380TF6cmobYp1+9aZ/JMzhtxFzvcu4q4iu9BvnRWnfY9J7LQJDCj[/tex]，又由于[tex=2.5x1.143]qulE2au0sCsC2RUF6/a3J6muFiZfq44W39c/ELCpKALI8lLIBAEhw5aGECGmh8jAet0vamZXm5qKAuZChoJRdw==[/tex]是1级矩阵，因此[tex=6.857x1.5]0idGSV3RW/tbV3escumNdCmdGMiK3qcee3NUzUm5bkJ7/r4WLsCHgfnvhOgwwVKwdF9c98TPacwtQw+fMUwB9SgxCqeBSQFS6MR4wVLqRGMPyBnN6W9yXpTikPcuN1aOUsc1rWSSNBgUz0MuK2XW3A8vv43JuIokwE8zY1JObMk=[/tex]，从而[tex=6.571x1.286]qulE2au0sCsC2RUF6/a3J6muFiZfq44W39c/ELCpKALI8lLIBAEhw5aGECGmh8jAHqBZhP/y0fYnnsZYna+P82eKAK4jXAKWsjCcML0MrhUULJ4VsChkOqBm1JbNrLO1vrtXjvP1fasmKM3dFYxkCw==[/tex]。由此得出，[tex=2.5x1.143]qulE2au0sCsC2RUF6/a3J6muFiZfq44W39c/ELCpKALI8lLIBAEhw5aGECGmh8jAet0vamZXm5qKAuZChoJRdw==[/tex]=0.充分性。设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的列向量组是[tex=6.357x1.0]qulE2au0sCsC2RUF6/a3Jzvc72/Ojlqjke3MkB3mi/ocVm1zbQVoiZ3n4s0Tg+DcUYb3pwbmP3EXwTF2glwNookvPqLjZ8WM2JwV9JFjusoGV3okOYIXPKl6YqwjmmTB[/tex]：由已知条件得[tex=15.071x1.429]N6j5tRzvjYiiwyJOGUqByT0JGy2TiZT6+5sYviG1mV2nf/lgphoBm7e3CfknF3Mb6fY07ROvnYthGA9wprfCVVrRzOYcxhDviyaal8Dadb/bw0HkkE1rVlJiJ9s5be7t4T6lLDSGORkmX1+HVqygSLf55DXigiHi7y0eCb7fWXCK+1NdozTlJyq/bkbfOcNKc6rlws6lvXuBHWsYJQukF8T4KN2v080SRWkz8JtNyl4=[/tex]。[tex=17.143x1.643]rJ4QosgCAuVuKB+Ku8/JYtn7KdQwzk4wIIf4GPOtRpZcP8Xy5Tl24iBWivEzvPzeB9J7DdyhhWFSXSrUQ+Xc60RaAtJXwOJBtSsijZa7kwWUeRYBOvJGCDNDuDHmWOl4xJLaRajd5fAVdchneMEFoaoqI+PoTEAl3UwR41bvACUKojTeOriUK4Vt1a05DJgqa6s3t3kzkTBMGSj20ozAQPqP8D3ryf78uQ5MjTkmQd4N46csuXi3y4SC3h6V2K9m4amV+u6ClMV1kfv8pYff7Iq+OA0KetYZgfq5tDaIk1V/3Wvd07SAZ4X7YHPh+ckOaaSAESwcX2hfY7HbEkjJSQzeJYioTAVPkv4L9/Gid64kW6OABDXnSVH5tiXeUJafWnKvOZxezgzdnpUAjIWiPA==[/tex][tex=13.571x1.286]7ZIOAbdo5S7NEYRrA81MwNfYcL3nLNspp7QuLzyqEXFBfb1iTJqmgrrRCWYNSWsNe6i9+0sgOi+sUUFRKST2qg==[/tex].因此[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是斜对称矩阵。

举一反三

内容

0
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，证明：如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]可对角化，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的伴随矩阵[tex=1.143x1.071]Z+TPszFO7LPa8KJ9E9RUwQ==[/tex]也可对角化。
1
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级可逆矩阵，证明：如果[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]可对角化，那么[tex=3.286x1.429]l5sF9EhDX0KUFjwu7SC5JQ==[/tex]都可对角化。
2
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵，证明：对任意[tex=3.429x1.143]irUvzt1wsrCMoSqyhzSP6A==[/tex]，有[tex=18.357x1.286]BeH+evXWqA87DrUQ87mj6ojNc+UedcSpxob7YvXfNnYH2g9EXmseUBVAuJVsCWq/qDaGCxms4rPlleV4wBQdaIqCBNhZ5v+duFBn4X16o5U=[/tex]。
3
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是一个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级正交矩阵，证明：如果[tex=2.643x1.357]xnNlsIp2wAAq+OkAnU/oIQ==[/tex]，且[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]是奇数，那么1是[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的一个特征值。
4
证明：如果数域[tex=0.857x1.0]eMszuSG5by5UfRZVROYp5A==[/tex]上的[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]满足[tex=6.643x1.429]oCpvLyGD8hllYcsM3cYSLWtvPKGckJqIibvm40exWHI=[/tex]，那么[tex=3.857x1.357]/ErxrDUA0p2I1qrW8TNM9Q==[/tex].