复旦大学谢启鸿高等代数每周一题[2021A06]参考解答

2021-11-06 18:09 作者:CharlesMa0606 0人读过 | 我要投稿

本文是本人给出的2021年复旦大学谢启鸿高等代数的每周一题[问题2021A06]的解答

题目来自于复旦大学谢启鸿教授在他的博客提供的每周一题练习

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

本文仅供学习交流，如有错误恳请指正！

[问题2021A06]若n阶实方阵P满足 $PP%5E%5Cprime%3DI_n$ ,则称P为正交阵.设S为n阶实反对称阵全体构成的集合, $T%3D%5C%7BP%7CP%E4%B8%BAn%E9%98%B6%E6%AD%A3%E4%BA%A4%E9%98%B5%E4%B8%94%E6%BB%A1%E8%B6%B3%20In%2BP%E5%8F%AF%E9%80%86%5C%7D$ .

(1)对任意的 $A%5Cin%20S$ , 由高代白皮书的例2.33可知 $I_n%2BA$ 可逆,定义 $%5Cvarphi%5Cleft(A%5Cright)%3D%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D$ , 证明: $%5Cvarphi$ 是从S到T的映射.

(2)对任意的 $P%5Cin%20T$ ,定义 $%5Cpsi%5Cleft(P%5Cright)%3D%5Cleft(In-P%5Cright)%5Cleft(In%2BP%5Cright)%5E%7B-1%7D$ ,证明: $%5Cpsi$ 是从T到S的映射.

(3)证明: $%5Cpsi%5Cvarphi%3DId_S%2C%5Cvarphi%5Cpsi%3DId_T$ ,其中 $Id_S%2CId_T$ 表示S,T上的恒等映射,即 $%5Cvarphi%2C%5Cpsi$ 实现了集合S与T之间的一一对应.

(4)设n阶实反对称阵

$A%3D%5Cleft(%5Cbegin%7Bmatrix%7D0%261%261%26%5Ccdots%261%5C%5C-1%260%261%26%5Ccdots%261%5C%5C-1%26-1%260%26%5Ccdots%261%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5C-1%26-1%26-1%26%5Ccdots%260%5C%5C%5Cend%7Bmatrix%7D%5Cright)$

试求 $%5Cvarphi%5Cleft(A%5Cright)%3D%5Cleft(In-A%5Cright)%5Cleft(In%2BA%5Cright)%5E%7B-1%7D$ .

解(1)对任意的 $A%5Cin%20S$ ，有：

$%5Cvarphi%5Cleft(A%5Cright)%3D%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%2C%5Cvarphi%5Cleft(A%5Cright)%5E%5Cprime%3D%5Cleft(%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cright)%5E%5Cprime%3D%5Cleft(I_n-A%5Cright)%5E%7B-1%7D%5Cleft(I_n%2BA%5Cright)$

从而

$%5Cvarphi%5Cleft(A%5Cright)%5E%5Cprime%5Cvarphi%5Cleft(A%5Cright)%3D%5Cleft(I_n-A%5Cright)%5E%7B-1%7D%5Cleft(I_n%2BA%5Cright)%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%3D%5Cleft(I_n-A%5Cright)%5E%7B-1%7D%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%3DI_n$

并且 $I_n%2B%5Cvarphi%5Cleft(A%5Cright)%3DI_n%2B%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%3D2%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D$ 是可逆阵.

从而 $%5Cvarphi%5Cleft(A%5Cright)%5Cin%20T%2C%5Cforall%20A%5Cin%20S$ ，因此 $%5Cvarphi$ 是从S到T的映射.

(2)对任意的 $P%5Cin%20T$ ，有：

$%5Cpsi%5Cleft(P%5Cright)%3D%5Cleft(I_n-P%5Cright)%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D%2C%5Cpsi%5Cleft(P%5Cright)%5E%5Cprime%3D%5Cleft(I_n%2BP%5E%5Cprime%5Cright)%5E%7B-1%7D%5Cleft(I_n-P%5E%5Cprime%5Cright)$

$%5Cpsi%5Cleft(P%5Cright)%2B%5Cpsi%5Cleft(P%5Cright)%5E%5Cprime%3D%5Cleft(I_n%2BP%5E%5Cprime%5Cright)%5E%7B-1%7D%5Cleft(I_n-P%5E%5Cprime%5Cright)%2B%5Cleft(I_n-P%5Cright)%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D$

$%3D%5Cleft(I_n%2BP%5E%5Cprime%5Cright)%5E%7B-1%7D%5Cleft(%5Cleft(I_n-P%5E%5Cprime%5Cright)%5Cleft(I_n%2BP%5Cright)%2B%5Cleft(I_n%2BP%5E%5Cprime%5Cright)%5Cleft(I_n-P%5Cright)%5Cright)%5Cleft(In%2BP%5Cright)%5E%7B-1%7D$

$%3D%5Cleft(In-P%5Cright)%5E%7B-1%7D%5Cleft(I_n%2BP-P%5E%5Cprime-I_n%2BI_n%2BP%5E%5Cprime-P-I_n%5Cright)%5Cleft(In%2BP%5Cright)%5E%7B-1%7D%3DO$ .

从而 $%5Cpsi%5Cleft(P%5Cright)%5Cin%20S%2C%5Cforall%20P%5Cin%20T$ ,因此 $%5Cpsi$ 是从T到S的映射.

(3)对于任意的 $A%5Cin%20S%2CP%5Cin%20T$ ,有

$%5Cpsi%5Cvarphi%5Cleft(A%5Cright)%3D%5Cleft(I_n-%5Cvarphi%5Cleft(A%5Cright)%5Cright)%5Cleft(I_n%2B%5Cvarphi%5Cleft(A%5Cright)%5Cright)%5E%7B-1%7D%3D%5Cleft(I_n-%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cright)%5Cleft(I_n%2B%5Cleft(I_n-A%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cright)%5E%7B-1%7D$ $%3D%5Cleft(%5Cleft(I_n%2BA%5Cright)-%5Cleft(I_n-A%5Cright)%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cleft%5B%5Cleft(%5Cleft(I_n%2BA%5Cright)%2B%5Cleft(I_n-A%5Cright)%5Cright)%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cright%5D%5E%7B-1%7D$ $%3D2A%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cleft(2%5Cleft(I_n%2BA%5Cright)%5E%7B-1%7D%5Cright)%5E%7B-1%7D%3DA.$

$%5Cvarphi%5Cpsi%5Cleft(P%5Cright)%3D%5Cleft(I_n-%5Cpsi%5Cleft(P%5Cright)%5Cright)%5Cleft(I_n%2B%5Cpsi%5Cleft(P%5Cright)%5Cright)%5E%7B-1%7D%3D%5Cleft(I_n-%5Cleft(In-P%5Cright)%5Cleft(In%2BP%5Cright)%5E%7B-1%7D%5Cright)%5Cleft(I_n%2B%5Cleft(In-P%5Cright)%5Cleft(In%2BP%5Cright)%5E%7B-1%7D%5Cright)%5E%7B-1%7D$ $%3D%5Cleft(%5Cleft(I_n%2BP%5Cright)-%5Cleft(I_n-P%5Cright)%5Cright)%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D%5Cleft%5B%5Cleft(%5Cleft(I_n%2BP%5Cright)%2B%5Cleft(I_n-P%5Cright)%5Cright)%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D%5Cright%5D%5E%7B-1%7D$

$%3D2P%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D%5Cleft(2%5Cleft(I_n%2BP%5Cright)%5E%7B-1%7D%5Cright)%5E%7B-1%7D%3DP.$

因此 $%5Cpsi%5Cvarphi%3DId_S%2C%5Cvarphi%5Cpsi%3DId_T$ ，即 $%5Cvarphi%2C%5Cpsi$ 实现了集合S与T之间的一一对应.

(4)用初等变换法求逆阵不难得到：

$%5Cleft(In%2BA%5Cright)%5E%7B-1%7D%3D%5Cfrac%7B1%7D%7B2%7D%5Cleft(%5Cbegin%7Bmatrix%7D1%26-1%260%26%5Ccdots%260%5C%5C0%261%26-1%26%5Ccdots%260%5C%5C0%260%261%26%5Ccdots%260%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5C1%260%260%26%5Ccdots%261%5C%5C%5Cend%7Bmatrix%7D%5Cright)$

从而

$%5Cvarphi%5Cleft(A%5Cright)%3D-%5Cfrac%7B1%7D%7B2%7D%5Cleft(%5Cbegin%7Bmatrix%7D-1%261%261%26%5Ccdots%261%5C%5C-1%26-1%261%26%5Ccdots%261%5C%5C-1%26-1%26-1%26%5Ccdots%261%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5C-1%26-1%26-1%26%5Ccdots%26-1%5C%5C%5Cend%7Bmatrix%7D%5Cright)%5Cleft(%5Cbegin%7Bmatrix%7D1%26-1%260%26%5Ccdots%260%5C%5C0%261%26-1%26%5Ccdots%260%5C%5C0%260%261%26%5Ccdots%260%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5C1%260%260%26%5Ccdots%261%5C%5C%5Cend%7Bmatrix%7D%5Cright)%3D%5Cleft(%5Cbegin%7Bmatrix%7D0%26-1%260%26%5Ccdots%260%5C%5C0%260%26-1%26%5Ccdots%260%5C%5C0%260%260%26%5Ccdots%260%5C%5C%5Cvdots%26%5Cvdots%26%5Cvdots%26%5C%20%26%5Cvdots%5C%5C1%260%260%26%5Ccdots%260%5C%5C%5Cend%7Bmatrix%7D%5Cright).$ $%5BQ.E.D%5D%0A%0A$