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6. For any n x n matrix A over C, there exists a positive semi- definite matrix H and a unitary matrix such that A = HU. If A is nonsingular, then H is positive definite and U and H are unique. (i) Find the polar decomposition for the invertible 3 x 3 matrix 1 0-4 A = 0 5 3). 4 -4 4 The positive definite matrix H is the unique square root of the positive definite matrix A* A and then U is defined by U = AH-¹. (ii) Apply the polar decomposition to the nonnormal matrix A 1 1 A -(¹) - 4 = (11) A* 01 which is an element of the Lie group SL(2, R).

6. For any n x n matrix A over C, there exists a positive semi- definite matrix H and a unitary matrix such that A = HU. If A is nonsingular, then H is positive definite and U and H are unique. (i) Find the polar decomposition for the invertible 3 x 3 matrix 1 0-4 A = 0 5 3). 4 -4 4 The positive definite matrix H is the unique square root of the positive definite matrix A* A and then U is defined by U = AH-¹. (ii) Apply the polar decomposition to the nonnormal matrix A 1 1 A -(¹) - 4 = (11) A* 01 which is an element of the Lie group SL(2, R).

Advanced Engineering Mathematics
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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6. For any n x n matrix A over C, there exists a positive semi- definite matrix H and a unitary matrix such that A = HU. If A is nonsingular, then H is positive definite and U and H are unique. (i) Find the polar decomposition for the invertible 3 x 3 matrix 1 0 -4 A = 0 5 4 -4 4 3 - The positive definite matrix H is the unique square root of the positive definite matrix A* A and then U is defined by U = AH-¹. (ii) Apply the polar decomposition to the nonnormal matrix A 1 A = -(64) (² A* = (1) 0 which is an element of the Lie group SL(2, R).
Transcribed Image Text:6. For any n x n matrix A over C, there exists a positive semi- definite matrix H and a unitary matrix such that A = HU. If A is nonsingular, then H is positive definite and U and H are unique. (i) Find the polar decomposition for the invertible 3 x 3 matrix 1 0 -4 A = 0 5 4 -4 4 3 - The positive definite matrix H is the unique square root of the positive definite matrix A* A and then U is defined by U = AH-¹. (ii) Apply the polar decomposition to the nonnormal matrix A 1 A = -(64) (² A* = (1) 0 which is an element of the Lie group SL(2, R).
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