Let n ≥ 2. Consider the reversing operator R: R" → Rn defined by R(a₁, 02, ..., an-1, an) = (an, an-1,..., a2, 0₁). We denote by E₁ and E-1 the eigenspaces of eigenvalues X = 1 and λ = -1, respec- tively. (1) (2) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.

Let n ≥ 2. Consider the reversing operator R: R" → Rn defined by R(a₁, 02, ..., an-1, an) = (an, an-1,..., a2, 0₁). We denote by E₁ and E-1 the eigenspaces of eigenvalues X = 1 and λ = -1, respec- tively. (1) (2) If n is even, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable. If n is odd, find a basis of E₁ and a basis of E-1, and determine whether or not R is diagonalizable.

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 41EQ

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