A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix. (a) Show that I − A is also idempotent. (b) Show that if A is invertible, then A = I. (c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix. (a) Show that I − A is also idempotent. (b) Show that if A is invertible, then A = I. (c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 24EQ
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A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix.
(a) Show that I − A is also idempotent.
(b) Show that if A is invertible, then A = I.
(c) Show that the only possible eigenvalues of A are 0 and 1. (Hint: Suppose x is an eigenvector with associated eigenvalue λ and then multiply x on the left by A twice.)
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