(a) If v is any nonzero vector perpendicular to u, show that v is an eigenvector of H and find the corresponding eigenvalue. How many times is that eigenvalue repeated? (b) Compute the sum of the diagonal entries of H and the sum of the eigenvalues of H.

(a) If v is any nonzero vector perpendicular to u, show that v is an eigenvector of H and find the corresponding eigenvalue. How many times is that eigenvalue repeated? (b) Compute the sum of the diagonal entries of H and the sum of the eigenvalues of H.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.1: Introduction To Eigenvalues And Eigenvectors

Problem 36EQ: Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has two...

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Question

Consider the n x n symmetric matrix H = I - 2uu^T, where u is a unit vector in Rⁿ, i.e., u^Tu = 1.

(a) If v is any nonzero vector perpendicular to u, show that v is an eigenvector of H and find the corresponding eigenvalue. How many times is that eigenvalue repeated?

(b) Compute the sum of the diagonal entries of H and the sum of the eigenvalues of H.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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