Author Archives: ccjou

每週問題 April 17, 2017

這是網友范智忠提供的問題。 Let and be matrices. If , show that .

Posted in pow 線性方程與矩陣代數, 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 April 3, 2017

找出所有的可交換矩陣。 Find all matrices commuting with , where is the matrix all elements of which are equal to .

Posted in pow 線性方程與矩陣代數, 每週問題 | Tagged | Leave a comment

每週問題 March 27, 2017

若對角矩陣有相異對角元與某個矩陣是可交換的,則該矩陣也是對角矩陣。 Prove the following statements. (a) Let , where are distinct. If , then is a diagonal matrix. (b) Let , where are distinct and nonzero, and be an matrix, where . If and , then .

Posted in pow 線性方程與矩陣代數, 每週問題 | Tagged | Leave a comment

每週問題 March 20, 2017

證明 為正規矩陣 (normal matrix) 是一個充分條件。 Let and be normal matrices such that , where denotes the column space of . Prove that is a normal matrix. Note that is a normal matrix if .

Posted in pow 二次型, 每週問題 | Tagged | 1 Comment

每週問題 March 13, 2017

任一正規矩陣 (normal matrix) 可表示為一個正規矩陣的平方。 Let be a normal matrix, i.e., . Prove that there exists a normal matrix such that .

Posted in pow 二次型, 每週問題 | Tagged | Leave a comment

每週問題 March 6, 2017

證明 Hermitian 矩陣的秩與跡數不等式。 Let be an nonzero Hermitian matrix. Prove that .

Posted in pow 二次型, 每週問題 | Tagged , , | Leave a comment

每週問題 February 27, 2017

利用相合 (congruence) 變換證明若 ,則 。 Let and be Hermitian matrices. We will write that if is positive definite. The inequality means that is positive definite. Prove that if , then .

Posted in pow 二次型, 每週問題 | Tagged , | Leave a comment

每週問題 February 20, 2017

證明三階旋轉矩陣的一個跡數恆等式。 Let be a real orthogonal matrix and . Prove that .

Posted in pow 特徵分析, 每週問題 | Tagged , , , | 1 Comment

每週問題 February 13, 2017

證明遍歷定理 (ergodic theorem)。 Let be a unitary matrix, i.e., . Prove that , where is the Hermitian projection matrix onto .

Posted in pow 內積空間, 每週問題 | Tagged , | 1 Comment