Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1, λ2, . . . , λk and corresponding multiplicities m1,m2, . . . ,mk. Suppose that β is a basis for V such that [T]β is an upper triangular matrix. Prove that the diagonal entries of [T]β are λ1, λ2, . . . , λk and that each λi occurs mi times (1≤i ≤k).
Let T be a linear operator on a finite-dimensional vector space V with the distinct eigenvalues λ1, λ2, . . . , λk and corresponding multiplicities m1,m2, . . . ,mk. Suppose that β is a basis for V such that [T]β is an upper triangular matrix. Prove that the diagonal entries of [T]β are λ1, λ2, . . . , λk and that each λi occurs mi times (1≤i ≤k).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 11AEXP
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Let T be a linear operator on a finite-dimensional
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