You are given that a 2 x 2 real matrix A has the following unit eigenvectors corresponding to a single eigenvalue X: 1 (1, -1), √2 1 X2 = (1,1) √2 1. Is A guaranteed to be symmetric, skew-symmetric, or neither? Is A guaranteed to be real, imginary, or neither? 2. Use the given information to write out a diagonalization of A that does not include any matrix inverses. 3. Write an expression for A as a summation of two projection matrices. 4. Is it guaranteed that Ax is a multiple of x for any 2-vector x? Explain why or why not. 5. Consider the ratio ||A²+1x|| ||Akx|| and assume that λ 0. What is the smallest value of k for which this ratio is equal to X? Explain whether this also holds true for larger values of k. X1 =

You are given that a 2 x 2 real matrix A has the following unit eigenvectors corresponding to a single eigenvalue X: 1 (1, -1), √2 1 X2 = (1,1) √2 1. Is A guaranteed to be symmetric, skew-symmetric, or neither? Is A guaranteed to be real, imginary, or neither? 2. Use the given information to write out a diagonalization of A that does not include any matrix inverses. 3. Write an expression for A as a summation of two projection matrices. 4. Is it guaranteed that Ax is a multiple of x for any 2-vector x? Explain why or why not. 5. Consider the ratio ||A²+1x|| ||Akx|| and assume that λ 0. What is the smallest value of k for which this ratio is equal to X? Explain whether this also holds true for larger values of k. X1 =

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

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