<br>设A的n个两两正交的特征向量为α<sub>1</sub>,α<sub>2</sub>,…,α<sub>n</sub>,其对应的特征值依次为<img src="/s/tiw/p3/UpLoadImage/2013-05-07/a8a0f3fb-333b-43a1-96ca-530ed82fd354.png" width="79" height="24">.<br>令<img src="/s/tiw/p3/UpLoadImage/2013-05-07/3f992864-a843-48e8-9e48-46cd4bc51a1f.png" width="147" height="48">,则<img src="/s/tiw/p3/UpLoadImage/2013-05-07/dded9276-dddd-41f1-afdc-671baeb39a19.png" width="76" height="24">是两两正交的单位向量.<br>记<img src="/s/tiw/p3/UpLoadImage/2013-05-07/0d03d322-372f-47e7-a1ee-054578933280.png" width="116" height="24">,即P是正交矩阵.从而有<br><img src="/s/tiw/p3/UpLoadImage/2013-05-07/c67dd1ce-c3bf-4099-a269-f8770d5d6ed3.jpg" width="325" height="28"><br>即<img src="/s/tiw/p3/UpLoadImage/2013-05-07/383550f3-a110-4acf-bcdd-79e29a54aeac.png" width="129" height="20">,故<img src="/s/tiw/p3/UpLoadImage/2013-05-07/d851e0fa-a44e-4aa0-8dd9-22ca9c00f67a.png" width="272" height="24">,即A是对称矩阵.
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