Let A be an n x n symmetric matrix, R be an n xm matrix such thatRTR = 1 (with m≤n), and let B = R¹ AR (an m × m matrix). The following propertieshold:(a) The eigenvalues of B interlace the eigenvalues of A....(0) If λι 5 λεAn are the eigenvalues of A and µ₁ ≤ µ₂ ≤... <μm are the2eigenvalues of B, and if λ = i, then there is an eigenvector v of B with eigenvaluei such that Rv is an eigenvector of A with eigenvalue i-

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Let A be an n x n symmetric matrix, R be an n xm matrix such that with m ≤n), and let B = R¹ AR (an m × m matrix). The following properties igenvalues of B interlace the eigenvalues of A. < 1₂ <...< An are the eigenvalues of A and µ₁ ≤ µ₂ <... m are the alues of B, and if λ = ₁, then there is an eigenvector v of B with eigenvalue h that Rv is an eigenvector of A with eigenvalue Xi-

Let A be an n x n symmetric matrix, R be an n xm matrix such that with m ≤n), and let B = R¹ AR (an m × m matrix). The following properties igenvalues of B interlace the eigenvalues of A. < 1₂ <...< An are the eigenvalues of A and µ₁ ≤ µ₂ <... m are the alues of B, and if λ = ₁, then there is an eigenvector v of B with eigenvalue h that Rv is an eigenvector of A with eigenvalue Xi-

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 24EQ

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