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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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2F7fe2df32-5594-46ca-92ab-8b557e01c223%2Fcdi074j_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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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