Tag Archives:

每週問題 June 5, 2017

證明 是不可逆矩陣的一個充分條件。 Let and be matrices, where is an odd number. Prove that if then at least one of the matrices and is singular. Advertisements

Posted in pow 向量空間, 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 October 24, 2016

對於任一複矩陣 , 總是成立嗎? Let be an complex matrix. Prove or disprove the following statements. (a) . (b) .

Posted in pow 向量空間, 每週問題 | Tagged | Leave a comment

每週問題 July 11, 2016

判定一個齊次系統的自由變數 (free variable) 的數目。 Suppose that is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that has at least one nonzero row. (a) Determine the fewest number of free variables that are possible. … Continue reading

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

每週問題 June 20, 2016

計算一個線性變換的秩。 Let be an orthonormal set in and be the cross product of and , i.e., . A linear transformation is defined by . Determine the rank of .

Posted in pow 線性變換, 每週問題 | Tagged , | Leave a comment

每週問題 May 23, 2016

這是關於矩陣秩的擾動問題。 Let and be matrices. If and , show that . In words, a perturbation of rank can change the rank by at most .

Posted in pow 向量空間, 每週問題 | Tagged | Leave a comment

每週問題 May 16, 2016

證明兩個矩陣的秩差的不等式。 Let and be matrices. Show that .

Posted in pow 向量空間, 每週問題 | Tagged | Leave a comment

每週問題 April 4, 2016

若 ,則 的最大秩是多少? Let be an matrix and . What is the maximum value of ?

Posted in pow 向量空間, 每週問題 | Tagged , , | Leave a comment

每週問題 January 18, 2016

證明兩個冪等 (idempotent) 矩陣相似的一個充要條件是它們有相同的秩。 Let and be idempotent matrices, i.e., and . Show that is similar to if and only if .

Posted in pow 特徵分析, 每週問題 | Tagged , , | Leave a comment

每週問題 January 11, 2016

證明秩─1線性算子的兩個性質。 Let be a linear operator of rank one. Prove the following statements. (a) There exists a unique scalar such that . (b) If , then is invertible, where is the identity transformation.

Posted in pow 線性變換, 每週問題 | Tagged , , | Leave a comment