Tag Archives: 秩

每週問題 June 5, 2017

Posted on 06/05/2017 by ccjou

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.

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每週問題 March 6, 2017

Posted on 03/06/2017 by ccjou

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

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每週問題 October 24, 2016

Posted on 10/24/2016 by ccjou

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) .

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每週問題 July 11, 2016

Posted on 07/12/2016 by ccjou

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 →

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每週問題 June 20, 2016

Posted on 06/20/2016 by ccjou

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 .

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每週問題 May 23, 2016

Posted on 05/23/2016 by ccjou

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 .

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每週問題 May 16, 2016

Posted on 05/16/2016 by ccjou

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

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每週問題 April 4, 2016

Posted on 03/31/2016 by ccjou

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

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每週問題 January 18, 2016

Posted on 01/18/2016 by ccjou

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 .

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每週問題 January 11, 2016

Posted on 01/11/2016 by ccjou

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.

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