Let A E Mnxn (F). Prove the following statements. (a) If {v₁,..., Un} is a basis for Fn that triangularizes LA, then the matrix Q = (v₁ Un) € Mnxn (F) is invertible, and Q-¹AQ is upper triangular. (b) Conversely, if QE Mnxn (F) is an invertible matrix such that Q-¹AQ is upper triangular, then the column vectors of Q form a basis for F" that triangularizes LA.

Let A E Mnxn (F). Prove the following statements. (a) If {v₁,..., Un} is a basis for Fn that triangularizes LA, then the matrix Q = (v₁ Un) € Mnxn (F) is invertible, and Q-¹AQ is upper triangular. (b) Conversely, if QE Mnxn (F) is an invertible matrix such that Q-¹AQ is upper triangular, then the column vectors of Q form a basis for F" that triangularizes LA.

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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