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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 A 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 A and then multiply x on the left by A twice.)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 A and then multiply x on the left by A twice.)
Transcribed Image Text: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 A and then multiply x on the left by A twice.)
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