Category Archives: pow 特徵分析

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).

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 .

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?

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 .

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 .

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

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 .

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.