Let A be a n × n matrix with real entries and let A1 = a + bi (where a and b are real and b# 0) be an eigenvalue of A. Let z1 = x + iy (where x and y both have real entries) be an eigenvector of A corresponding to A1, and let z2 = x – iy. (a) Explain why z1 and z2 must be linearly independent as vectors in C". (b) Show that y # 0 and that x and y are linearly independent as vectors in R".

Let A be a n × n matrix with real entries and let A1 = a + bi (where a and b are real and b# 0) be an eigenvalue of A. Let z1 = x + iy (where x and y both have real entries) be an eigenvector of A corresponding to A1, and let z2 = x – iy. (a) Explain why z1 and z2 must be linearly independent as vectors in C". (b) Show that y # 0 and that x and y are linearly independent as vectors in R".

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 27EQ

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