復(fù)旦大學(xué)謝啟鴻高等代數(shù)每周一題[2021A14]參考解答

2021-12-19 20:45 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2021A14]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(cuò)誤懇請(qǐng)指正！

[問題2021A14]設(shè)V是n維復(fù)線性空間， $%5Cvarphi$ 是V上的線性變換.證明：可將V看成是2n維實(shí)線性空間 $V_0$ ， $%5Cvarphi$ 看成是 $V_0$ 上的實(shí)線性變換 $%5Cvarphi_0$ ，并且 $det%5Cleft(%5Cvarphi_0%5Cright)%3D%5Cleft%7Cdet%5Cleft(%5Cvarphi%5Cright)%5Cright%7C%5E2$ ，其中 $%5Cleft%7C%5C%20%5Ccdot%20%5C%20%5Cright%7C$ 表示復(fù)數(shù)的模長.

注? 因?yàn)榫€性變換在不同基下的表示矩陣是相似的，而且矩陣的行列式在相似關(guān)系下不改變，所以線性變換的行列式定義為它的任一表示矩陣的行列式.

證明? 取V基 $%5C%7Be_1%2C%5Ccdots%2Ce_n%5C%7D$ ，我們斷言可以將V看作2n維實(shí)線性空間 $V_0$ ，其中 $V_0%3DL%5Cleft(e_1%2C%5Ccdots%2Ce_n%2Cie_1%2C%5Ccdots%2Cie_n%5Cright)$ ，注意到在實(shí)線性空間 $V_0$ 中 $%5C%7Be_1%2C%5Ccdots%2Ce_n%2Cie_1%2C%5Ccdots%2Cie_n%5C%7D$ 線性無關(guān)，按定義證明:

設(shè)存在 $a_k%2Cb_k%5Cin%5Cmathbb%7BR%7D%2Cs.t.a_1e_1%2B%5Ccdots%2Ba_ne_n%2Bb_1%5Cleft(ie_1%5Cright)%2B%5Ccdots%2Bb_n%5Cleft(ie_n%5Cright)%3D0$ ，則 $z_1e_1%2B%5Ccdots%2Bz_ne_n%3D0$ ,其中 $z_k%3Da_k%2Bb_ki%5Cin%5Cmathbb%7BC%7D$ ，于是由 $%5C%7Be_1%2C%5Ccdots%2Ce_n%5C%7D$ 是復(fù)線性空間V基，于是 $z_k%3D0%5CLeftrightarrow%20a_k%3Db_k%3D0$ ，即 $%5C%7Be_1%2C%5Ccdots%2Ce_n%2Cie_1%2C%5Ccdots%2Cie_n%5C%7D$ 是 $V_0$ 基.

對(duì)任意 $%5Calpha%3Dz_1e_1%2B%5Ccdots%2Bz_ne_n%5Cin%20V%2Cz_k%3Da_k%2Bb_ki%5Cin%5Cmathbb%7BC%7D%2Ca_i%2Cb_i%5Cin%5Cmathbb%7BR%7D$ 則有 $%5Calpha%3Da_1e_1%2B%5Ccdots%2Ba_ne_n%2Bb_1%5Cleft(ie_1%5Cright)%2B%5Ccdots%2Bb_n%5Cleft(ie_n%5Cright)%5Cin%20V_0$ ，于是 $V%5Csubseteq%20V_0$ ，同理可證 $V_0%5Csubseteq%20V$ .從而有 $V%3DV_0$ .

設(shè) $%5Cvarphi$ 的表示陣為 $A%3D%5Cleft(a_%7Bij%7D%5Cright)%5Cin%20M_n%5Cleft(%5Cmathbb%7BC%7D%5Cright)$ ，則由表示陣的定義可知

$%5Cleft(%5Cvarphi%5Cleft(e_1%5Cright)%2C%5Ccdots%2C%5Cvarphi%5Cleft(e_n%5Cright)%5Cright)%3D%5Cleft(e_1%2C%5Ccdots%2Ce_n%5Cright)A$

更具體地，

$%5Cvarphi%5Cleft(e_k%5Cright)%3D%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bjk%7De_j%7D%3D%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7BRe%5Cleft(a_%7Bjk%7D%5Cright)e_j%7D%2B%5Csum_%7Bj%3D1%7D%5E%7Bn%7DIm%5Cleft(a_%7Bjk%7D%5Cright)%5Cleft(ie_j%5Cright)$

$%5Cvarphi%5Cleft(ie_k%5Cright)%3Di%5Cvarphi%5Cleft(e_k%5Cright)%3Di%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7Ba_%7Bjk%7De_j%7D%3D%5Csum_%7Bj%3D1%7D%5E%7Bn%7DRe%5Cleft(a_%7Bjk%7D%5Cright)%5Cleft(ie_j%5Cright)-%5Csum_%7Bj%3D1%7D%5E%7Bn%7D%7BIm%5Cleft(a_%7Bjk%7D%5Cright)e_j%7D$

于是我們定義 $%5Cvarphi_0%3AV_0%5Crightarrow%20V_0%2Cs.t.%5Cvarphi_0%5Cleft(e_k%5Cright)%3D%5Cvarphi%5Cleft(e_k%5Cright)%2C%5Cvarphi_0%5Cleft(ie_k%5Cright)%3D%5Cvarphi%5Cleft(ie_k%5Cright)$ ，從而 $%5Cvarphi$ 能看成是 $V_0$ 上的實(shí)線性變換 $%5Cvarphi_0$ .由表示陣的定義，我們有：

$%5Cleft(%5Cvarphi%5Cleft(e_1%5Cright)%2C%5Ccdots%2C%5Cvarphi%5Cleft(e_n%5Cright)%2C%5Cvarphi%5Cleft(ie_1%5Cright)%2C%5Ccdots%2C%5Cvarphi%5Cleft(ie_n%5Cright)%5Cright)%3D%5Cleft(e_1%2C%5Ccdots%2Ce_n%2Cie_1%2C%5Ccdots%2Cie_n%5Cright)B%2CB%3D%5Cleft(%5Cbegin%7Bmatrix%7DRe%5Cleft(A%5Cright)%26-Im%5Cleft(A%5Cright)%5C%5CIm%5Cleft(A%5Cright)%26Re%5Cleft(A%5Cright)%5C%5C%5Cend%7Bmatrix%7D%5Cright)$

由高代白皮書例2.60可知，或者利用初等變換法，有

$det%5Cleft(%5Cvarphi_0%5Cright)%3Ddet%5Cleft(B%5Cright)%3Ddet%5Cleft(Re%5Cleft(A%5Cright)%2BiIm%5Cleft(A%5Cright)%5Cright)det%5Cleft(Re%5Cleft(A%5Cright)-iIm%5Cleft(A%5Cright)%5Cright)%3D%5Cleft%7Cdet%5Cleft(A%5Cright)%5Cright%7C%5E2$