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.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A is n*n matrix

2. Let A € 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 Q = 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.

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