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復(fù)旦大學(xué)謝啟鴻高等代數(shù)每周一題[2021A03]參考解答

2021-10-19 21:54 作者:CharlesMa0606 0人讀過 | 我要投稿

本文是本人給出的2021年復(fù)旦大學(xué)謝啟鴻高等代數(shù)的每周一題[問題2021A03]的解答

題目來自于復(fù)旦大學(xué)謝啟鴻教授在他的博客提供的每周一題練習(xí)

（鏈接：https://www.cnblogs.com/torsor/p/15329047.html）

本文僅供學(xué)習(xí)交流，如有錯誤懇請指正！

[問題2021A03]設(shè)多項式 $f%5Cleft(x%5Cright)%3Dx%5En%2Ba_1x%5E%7Bn-1%7D%2B%5Ccdots%2Ba_%7Bn-1%7Dx%2Ba_n$ ， $A$ 是 $f(x)$ 的友陣:

? $A%3D%5Cleft(%5Cbegin%7Bmatrix%7D%5C%20%26%5C%20%26%5C%20%26%5C%20%26-a_n%5C%5C1%26%5C%20%26%5C%20%26%5C%20%26-a_%7Bn-1%7D%5C%5C%5C%20%26%5Cddots%26%5C%20%26%5C%20%26%5Cvdots%5C%5C%5C%20%26%5C%20%26%5Cddots%26%5C%20%26%5Cvdots%5C%5C%5C%20%26%5C%20%26%5C%20%261%26-a_1%5C%5C%5Cend%7Bmatrix%7D%5Cright)$

設(shè) $n$ 階矩陣 $X$ 滿足 $XA%3DA%5E%5Cprime%20X$ ，求證： $X$ 是對稱陣.

解?? $%E8%AE%BEX%3D%5Cleft(x_%7Bij%7D%5Cright)_%7Bn%5Ctimes%20n%7D%2C%E5%88%86%E5%88%AB%E8%80%83%E8%99%91%E7%AD%89%E5%8F%B7%E4%B8%A4%E8%BE%B9%E7%9A%84%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95%E7%9A%84%E7%BB%93%E6%9E%9C%EF%BC%8C%E6%9C%89%EF%BC%9A$

$XA%3D%5Cleft(%5Cbegin%7Bmatrix%7Dx_%7B12%7D%26x_%7B13%7D%26%5Ccdots%26x_%7B1n%7D%26-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7B1i%7Da_%7Bn-i%2B1%7D%7D%5C%5Cx_%7B22%7D%26x_%7B23%7D%26%5Ccdots%26x_%7B2n%7D%26-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7B2i%7Da_%7Bn-i%2B1%7D%7D%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%26%5Cvdots%5C%5Cx_%7Bn2%7D%26x_%7Bn3%7D%26%5Ccdots%26x_%7Bnn%7D%26-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7Bni%7Da_%7Bn-i%2B1%7D%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright)$

$A%5E%5Cprime%20X%3D%5Cleft(%5Cbegin%7Bmatrix%7Dx_%7B21%7D%26x_%7B22%7D%26%5Ccdots%26x_%7B2n%7D%5C%5Cx_%7B31%7D%26x_%7B32%7D%26%5Ccdots%26x_%7B3n%7D%5C%5C%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5Cx_%7Bn1%7D%26x_%7Bn2%7D%26%5Ccdots%26x_%7Bnn%7D%5C%5C-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7Bi1%7Da_%7Bn-i%2B1%7D%7D%26-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7Bi2%7Da_%7Bn-i%2B1%7D%7D%26%5Ccdots%26-%5Csum_%7Bi%3D1%7D%5E%7Bn%7D%7Bx_%7Bin%7Da_%7Bn-i%2B1%7D%7D%5C%5C%5Cend%7Bmatrix%7D%5Cright)$