Category Archives: pow 特徵分析

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 . Advertisements

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每週問題 September 12, 2016

證明可交換矩陣的一個充要條件。 Let and be matrices. Suppose that the eigenvectors of span and have distinct eigenvalues. Show that if and only if and have the same set of eigenvectors (with possibly different eigenvalues).

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每週問題 April 25, 2016

計算一個線性變換的跡數、行列式、特徵值與特徵向量。 Let be the vector space spanned by functions and . (a) Find the trace and determinant of the linear transformation from to . (b) Find the eigenvalues and corresponding eigenvectors of .

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每週問題 February 22, 2016

改變三個矩陣乘積順序,特徵值是否改變? Let , , and be matrices. (a) Is it true that , , and have the same eigenvalues? (b) Is it true that and have the same eigenvalues?

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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 .

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每週問題 January 4, 2016

計算可交換矩陣構成的分塊矩陣的特徵值。 Let and be matrix. If , find the eigenvalues of .

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每週問題 December 28, 2015

計算 的特徵值。 Let be an matrix. Find the eigenvalues of in terms of those of .

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每週問題 December 21, 2015

這是計算一線性變換的特徵值與特徵向量。 Let be an matrix, and be the linear transformation defined by . For , find the eigenvalues and corresponding eigenvectors of .

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每週問題 November 9, 2015

這是廣義特徵值 (generalized eigenvalue) 問題。 Let and be matrices. If is nonsingular, show that there is a complex scalar such that is singular.

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每週問題 October 19, 2015

這是可交換矩陣的多項式表達問題。 Let and be matrices such that . If has distinct eigenvalues, show that can be expressed uniquely as a polynomial in with degree no more than .

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