Prove (b) of Theorem 6.15.Theorem 6.15. Let V be an inner product space, and let T be a normal operator on V. Then the following statements are true. (a) ||T(x)||= ||T∗(x)||for all x ∈V. (b) T – c|is normal for every c ∈F . (c) f x is an eigenvector of T, then x is also an eigenvector of T∗. In fact, if T(x) = λx, then T∗(x) = λx. (d) If λ1 and λ2 are distinct eigenvalues of Twith corresponding eigenvectors x1 and x2, then x1 and x2are orthogonal.

Prove (b) of Theorem 6.15.Theorem 6.15. Let V be an inner product space, and let T be a normal operator on V. Then the following statements are true. (a) ||T(x)||= ||T∗(x)||for all x ∈V. (b) T – c|is normal for every c ∈F . (c) f x is an eigenvector of T, then x is also an eigenvector of T∗. In fact, if T(x) = λx, then T∗(x) = λx. (d) If λ1 and λ2 are distinct eigenvalues of Twith corresponding eigenvectors x1 and x2, then x1 and x2are orthogonal.

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 32EQ

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