设n阶矩阵A的[tex=1.0x1.286]cX62TJPpzV+1vN2v5FuSOg==[/tex]个元素为1，试求可逆矩阵P，使[tex=3.357x1.286]c0UF8VG1YfBAkdtQdtGjUA==[/tex]为对角阵，并写出与A相似的对角阵。

2022-06-29

设n阶矩阵A的[tex=1.0x1.286]cX62TJPpzV+1vN2v5FuSOg==[/tex]个元素为1，试求可逆矩阵P，使[tex=3.357x1.286]c0UF8VG1YfBAkdtQdtGjUA==[/tex]为对角阵，并写出与A相似的对角阵。

答案：

由于[tex=10.286x4.643]No14tepOrgpLFcwU7iwUQf9x6b9WbMdVE4ZNhcBb/WJV5Bkm2cyC2R874BN7FzpehxwKztjp+8CxmwZtzztNcErVtyjbbUq2C+LXNX/4KB/K5Mcy6xfmdtJOnL6SILorUSARpFFOILc2kNAXxng08El34aMylPfbQDTBS6eH48+IbQol5X7xkUSBLAj76z76azdoSnJOYhQmvRDXV6Leuw==[/tex]，则A的特征方程为[tex=16.571x5.214]GbD4oVyj8A7x92wAA0ZCxHlr65P5mBoNre1aDci6LvnySxxQ9dYd5BBX7BFFEqEwfd2mmqVkjS/ABFIkzh+vwLjnyfX2nyft1ZJVck3SkY/qa92goYqB6Wl7o05K+MNm5c1cbbpaWw6LCQFDoin+NGBp2UBtgv8WtDRCbcsD9dBRSo6k+bq4BJ6CgqKemA60vKBjAyiIuBoITgSQPiPyAZrdyI0sU2maYtvNbx784G5ATYSWLtqJNQG5gdiorH33[/tex][tex=11.857x5.214]mqOKs44jrJgbK/+3e5yEgbGT4YyseLsTzkg6aCnELsetEKWgK9ZbiBjZ//0hz7KlzGoQYBWrkjaV0HggClAIZY71I+6Ryh7y62A3JwrYQgicKGxezOWBoQKDzW3IDEfB16tlK948rpbO23yiQMhiKPIg/pmHxs3smWSL3grLNAuSReSTaZLzNyYVg6xw2fGyAeQAuRybTKQMoew5wBdpTTXV4pvax8rj5ZNilU9uaXaZR5eo32AjpQQ41xZCRUOC[/tex]             [tex=15.643x5.214]AbZlddJeOJ6dRZqHO8hVAQuiEC0BfqxbJyfyTf+yXqIUKL5s1HpudX/7WWhsZGR2OrFK5IKVeNugcV7FmSkkO++ufiTyX4GNfBzj7p/hpHk89QCyQwEBZMujIgJLlWJC3p0OOeOTqaH9W3Umx9WoCmLQpOb33o8cR/mmrZylyYKEYYhJTXLb9WFAGkzsDurubcvDlBtJCFrTiZbHKOP+v8Z4Lvf4XxZTmTcO2YJn/6XWy9QffGF1iP74KVMnfdId[/tex][tex=10.571x4.786]/IYMWYgk9XjWjehxJBoohvCYOIHg3sXDVYht/qXN4plCDw/AJUAPogrbZ0r04NkXXY6Ps6a3mhLyM46kqwJI84p1sHrBlPpwQOU8Ny+RTdzwNmh1EG/lAiufN5UjjUp3db5XWQYiVVpDPt3TmpxWLsp/47Hi6YCyMySFlrF8uQyTMs+VYeyUvS0KSeaTc6wTqH94XRYWWCqWny9LudrqpSJqjg0fslxGHrXeqnJoStU=[/tex]              [tex=5.857x1.357]Cj3AHEJKJ9/oMwwCnhjGgdrWWolH/8p8fj++jzKYIMw=[/tex]故A的全部特征值为[tex=6.786x1.286]/Mps0mDC630tPRZL0gD9LXOTf0nNY6KlrWjtZJGZayY=[/tex]，[tex=2.357x1.214]zmfOUoL065VVS6Puv4Lz0Q==[/tex].当[tex=2.786x1.286]YACayYCTRCFAlC3l1njk5tX2edEQTg36bWxOV7BazEo=[/tex]时，由[tex=6.286x1.357]sUx8Ay2hegeQlGhEdkp/YQ+/xEE95lrY8RXJ6TivvMjh1CIuyKjwb3+AbGlhevJX[/tex]，即[tex=10.929x5.786]rwMhqGKFQ+j3l2qMx/grPtWlxL8bBYrkAJcYjNncJFR0EVJq0lJ4l96MdJOwb7QGWGGUjGEcoX2G4nGpIkL1JE1h8qe1bselSPU1bT/fH67SqifIhjZfePWyzLlNrXPbeq+wrcNLDv2eLxvJmov8vMAsWon5Ettefq9JOcN9QA+weToG7Q9UWXPyxIkB4wIyWPoPaBkmKZNdq2UXbJFLPEacU4gqDgMQklwKc57gABU=[/tex][tex=8.143x5.929]jyVOORWehIbTNQvvtYroWnkBO03WCvjlRxW7hUFsdeHQdmW65ODprM0cdscxihHbrseeLh3F8xL9hw3ecAM0IxrRStZGdYeW3ANSfz35uhA9KCo6MwRbcZifXn5zoNudhQPHJ6rvGeduaIc1g/l/LnZZ6JQdFNJ2SiIYY2qXPuOs29CKzetKF9ZEn3AQy/Y8ALZ6v17qTuI8Xe/yJUI+GpDAckv6L67zxBpGfPrpnCw=[/tex]可得线性无关的特征向量[tex=10.286x1.5]c35uAiFm1RG42uhRSd5WtOzHQum8iVItweR0eHhkzDTGF7E+r1Lj/4rChv19Jq80[/tex][tex=10.286x1.5]N8Snas7od01jjX+op1+Z6CHx7fJb8ccW8893ffbf5qQLWj0UsGsrQ3TribuZbzBA[/tex][tex=12.857x1.5]LOOQ5lADyW0MO7aLMxVKOG9/E1HWciFsHoq9o9oDpdap+55dHzg3ekN1FO50OD2H[/tex]。当[tex=2.929x1.286]CZaP3VIeLSxwsAK6eRcEF99Sva9Uva17dzqxHa6u5ls=[/tex]时，由[tex=6.286x1.357]sUx8Ay2hegeQlGhEdkp/YQ0iHFOPez/k/S7Sc6sl5b53QWFBr0Qo5DQpzTtedoR4[/tex]，即[tex=12.714x5.214]rwMhqGKFQ+j3l2qMx/grPnNhUiAhBza2IOFhOpkLaCWzPrOwMYKK0FfNNQzSzIXCEV+hVpyUvHPzeiOtw9dZCieVs6eXzE6/ObMQ7B0twwomxZzSVM3rvnx2n8T/k0yUbj1+aCtVX7GASiSyP8EaSwHrYC+rnLW7yAf+QBz2ENJeoc5zto0HqpMXcWsI72X29rXwHCSqPK9uZqgte0Nq0g==[/tex][tex=8.143x4.643]jyVOORWehIbTNQvvtYroWnkBO03WCvjlRxW7hUFsdeHCaYXCCxC9N9MQD92LkmvGrnmnQwh6ZOqXH4trtla5ScG74HaVvrKCIgZanD7VkO4fsB/61fmlZMYVFYidiNJE+Lyu5M0qZkpXgQQgMZlVHFMDwdMKIOZQN1i5iMSXn9xtdM09iDV8eyOll8NwIl2Q[/tex]可得线性无关的特征向量[tex=9.286x1.5]YmV8Ag9QeuqI/CKNipEUrtLJyrH9r8qL7dUZjDONqdrFY0gj+ZymH7NBUZ8Yw0BM[/tex]由于A的每个特征值对应的特征向量线性无关的最大个数等于该特征值的重数，根据定理5.9可知A可以对角化，且存在可逆矩阵[tex=21.571x6.071]FNJQJVcN8HABLQHRxcENsNmpI1N5aHgElgA2WITwyPCk9sFTrTW9bUpdNZu+d1haT6hVKDp0j0J7JKmuQFk7Dypf/TcsbJRHU5+lyBPbn6bVHIFkuqGHcI4ezbhviLy1KQBse7xNHdbJDq76bEpI8zUujv6eIpczmGgz4hkq1mHNIETkC1jUtMwueam7EvxO/kWHBQDzgxO6DKThqPejpZZW9/iXpdv/i4HOn1sCnxfNYUzQ+Uzp/38o9akfGaYuawR0ra9cTOXIX0we0luGxYkBAb3mPE82UZjMlfhCjSVova0jW20VCMt3E9Bq5J5OLrWH2yyocnEOh4P76nzNxQ==[/tex]和对角矩阵[tex=10.0x1.357]+HYdo7hXvgwQG3BfDH8z8YX7uJ4ZtMqX17bchK5tbCNpuxVdnK1QxsaCNC3Sdhj7u1SRvl0hK6S7hGsNmgL4Cg==[/tex]使得[tex=5.286x1.286]rhGl7XA/qs6+9yphb1bfqlKSCNNHfnRAn8fewT7Oq/c=[/tex]。

举一反三

内容

0
矩阵[tex=6.786x2.786]lRsc+7xS9mVs48x3DLiOg2asEhVTfp50N5p5QSXNLqmieFRXSBTqqJhWuRQsNSjt4wIkX2rw/c/q57ZeTyN63g==[/tex]是否对角化? 若可对角化，试求可逆矩阵 [tex=0.929x1.214]4M4JO+cg8PL6vWL6afoCdg==[/tex] 使 [tex=3.143x1.214]W4jiGACeVytyGqwMmeXGeQ==[/tex]为对角阵。
1
下列矩阵是否与对角矩阵相似？若相似于对角矩阵, 求 [tex=0.714x1.286]BMKsEVFNvpiLV0UsqDFXCw==[/tex] 使 [tex=3.214x1.286]1sFdXzzqHCc4qLXdM8ki4Toicy90dsQBVNzF4LK8JX8=[/tex] 为对角矩阵.[p=align:center][tex=5.357x2.786]jcCMHflCR8OS9TosV6N5vFWjToaaWqOOGqoRSEmRakI8euajTYJW+cFHO0sg+D0a+NjWo5p5K3fsrlwkGSJ1tg==[/tex]
2
设矩阵 [tex=8.286x3.5]3BT1BgBZQ5uJXxD5dg+w2xTK9vs7uQA24QN5Zc8+9oZiDzNOUSILOEfV5fKHPQSqCCIHe8KxDVRuumO5bTFF2eJ9JdFPwlS6oajtAUt55jzcsa2EAGYg04XF8MTN1vyu[/tex] 问矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 可否相似对角化? 若能相似对角化, 则求正交阵 [tex=0.643x1.0]WUJ/JHItsc3Bqx1WYNJcrg==[/tex], 使 [tex=3.143x1.214]TB4LkDRgOTHK2s8VISumoQ==[/tex] 为对角阵.
3
设[tex=7.429x3.643]hB8sGfF8hpZRTKdvt1J/eDCVPMEOyrPt798Vn+R5Qt117L7Yi7lueww+a8BaqGsNcRuqNs/GWunY+/HuAUPFn6fF/f5NN5uuuuaDhM/IyzNkCBOKivfJp15cQlXwdUBH[/tex]，求可逆矩阵[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]，使[tex=2.286x1.143]4Yac+UFr8uz07yqJVSbibA==[/tex]为对角阵．
4
下列矩阵是否与对角矩阵相似？若相似于对角矩阵, 求 [tex=0.714x1.286]BMKsEVFNvpiLV0UsqDFXCw==[/tex] 使 [tex=3.214x1.286]1sFdXzzqHCc4qLXdM8ki4Toicy90dsQBVNzF4LK8JX8=[/tex] 为对角矩阵.[p=align:center][tex=7.857x3.929]jcCMHflCR8OS9TosV6N5vNF0Ht7EtBbeDLHluhYHejByT6aLXCSmkH2ygZWvIirihyGjHbIbgSgvxDJ9x8yJnPuBRfs11OT98H/vffeQswT2hcIZl8u2tnvwMLBc2tWO[/tex]