The 2×2 real symmetric matrix A has two distinct eigenvalues λ1 and λ2.(a) If v1 =(a,b) is an eigenvector of A corresponding to the eigenvalue λ1, determine an eigenvector corresponding to λ2,and hence ﬁnd an orthogonal matrix S such that ST AS=diag(λ1,λ2).(b) Use your result from part (a) to ﬁnd A. [Your answer will involve λ1,λ2,a, and b.]

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Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.4: The Singular Value Decomposition

Problem 26EQ

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The 2×2 real symmetric matrix A has two distinct eigenvalues λ₁ and λ₂.

(a) If v₁ =(a,b) is an eigenvector of A corresponding to the eigenvalue λ₁, determine an eigenvector corresponding to λ₂,and hence ﬁnd an orthogonal matrix S such that S^T AS=diag(λ₁,λ₂).

(b) Use your result from part (a) to ﬁnd A. [Your answer will involve λ₁,λ₂,a, and b.]

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