Q3. A matrix A e C™X" is called skew-Hermitian if A* = - A. Show that skew-Hermitian matrices have only purely imaginary eigenvalues, i.e., any eigenvalue of a skew-Hermitian matrix has the form ) = ai with a E R. Hint: Consider the Schur decomposition, and see what it means that A* = -A.

Q3. A matrix A e C™X" is called skew-Hermitian if A* = - A. Show that skew-Hermitian matrices have only purely imaginary eigenvalues, i.e., any eigenvalue of a skew-Hermitian matrix has the form ) = ai with a E R. Hint: Consider the Schur decomposition, and see what it means that A* = -A.

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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Q3. A matrix A e C™×" is called skew-Hermitian if A* = -A.
Show that skew-Hermitian matrices have only purely imaginary eigenvalues, i.e., any
eigenvalue of a skew-Hermitian matrix has the form ) = ai with a ER.
Hint: Consider the Schur decomposition, and see what it means that A* = -A.

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