Let A € Mnxn (F). Prove the following statements. (a) If {v₁,..., Un} is a basis for F" that triangularizes LA, then the matrix Q = (v₁ Un) € Mnxn (F) is invertible, and Q¹AQ 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.
Let A € Mnxn (F). Prove the following statements. (a) If {v₁,..., Un} is a basis for F" that triangularizes LA, then the matrix Q = (v₁ Un) € Mnxn (F) is invertible, and Q¹AQ 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.
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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Transcribed Image Text:Let A € Mnxn (F). Prove the following statements.
(a) If {v₁,..., Un} is a basis for F" that triangularizes LA, then the matrix
Q = (v₁
Un) € Mnxn (F)
is invertible, and Q¹AQ 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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