Category Archives: pow 行列式

每週問題 June 26, 2017

對於秩-1方陣 ,證明 。 Let be an matrix and . Prove that . Advertisements

Posted in pow 行列式, 每週問題 | Tagged , | 2 Comments

每週問題 May 22, 2017

以伴隨矩陣的行列式表達分塊矩陣的行列式。 Suppose is , is , is , and is a number. Prove that .

Posted in pow 行列式, 每週問題 | Tagged , | 1 Comment

每週問題 May 15, 2017

反對稱矩陣的伴隨矩陣 (adjugate) 是對稱或反對稱矩陣。 Let be an skew-symmetric matrix. Prove that is a symmetric matrix for odd and a skew-symmetric matrix for even .

Posted in pow 行列式, 每週問題 | Tagged , , | Leave a comment

每週問題 April 10, 2017

計算 的伴隨矩陣。 Let and be -dimensional column vectors. Prove that .

Posted in pow 行列式, 每週問題 | Tagged , | 2 Comments

每週問題 November 7, 2016

若一個 Hermitian 矩陣的主對角元為其特徵值,則此矩陣是對角矩陣。 Let be an Hermitian matrix whose eigenvalues, including multiple appearances, are the diagonal elements , . Prove that is diagonal.

Posted in pow 行列式, 每週問題 | Tagged , | 3 Comments

每週問題 October 5, 2015

證明 Jacobi 行列式恆等式。 Let be an nonsingular matrix, where is . Denote the adjugate of by , where is . Prove the Jacobi identity .

Posted in pow 行列式, 每週問題 | Tagged , | Leave a comment

每週問題 September 14, 2015

計算 的導數。 Let be an matrix, where each entry is a differentiable function of . Prove that , where is identical to except that the entries in the column are replaced by their derivatives, i.e., if , if .

Posted in pow 行列式, 每週問題 | Tagged | Leave a comment

每週問題 May 25, 2015

這是關於分塊矩陣行列式的計算問題。 Let , where and are square matrices of order and , respectively. Let be an matrix and be an matrix. Prove the following identities. (a) . (b) .

Posted in pow 行列式, 每週問題 | Tagged , | Leave a comment

每週問題 April 27, 2015

這是利用行列式證明一特殊矩陣型態必定可逆的問題。 For any matrix , show that there exists a matrix such that is nonsingular.

Posted in pow 行列式, 每週問題 | Tagged | Leave a comment

每週問題 January 19, 2015

如果 和 的所有元為整數,則 有甚麼性質? Let be an matrix. If all entries of and are integers, show that .

Posted in pow 行列式, 每週問題 | Tagged | Leave a comment