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Let A be an m x n matrix over C. The Moore-Penrose pseudo inverse matrix A+ is the unique n x m matrix which satisfies AA+ A = A, A* AA+ = A*, (AA*)* = AA*, (A+A)* = A* A. %3D We also have that x = A+b is the shortest length least square solution to the problem Ax = b. (i) Show that if (A* A)-1 exists, then A+ = (A* A)-'A*. (ii) Let %3D 1 3 A = | 2 4 3 5 Find the Moore-Penrose matrix inverse A+ of A.

Let A be an m x n matrix over C. The Moore-Penrose pseudo inverse matrix A+ is the unique n x m matrix which satisfies AA+ A = A, A* AA+ = A*, (AA*)* = AA*, (A+A)* = A* A. %3D We also have that x = A+b is the shortest length least square solution to the problem Ax = b. (i) Show that if (A* A)-1 exists, then A+ = (A* A)-'A*. (ii) Let %3D 1 3 A = | 2 4 3 5 Find the Moore-Penrose matrix inverse A+ of A.

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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Let A be an m × n matrix over C. The Moore-Penrose pseudo inverse matrix A+ is the unique n x m matrix which satisfies AA* A = A, A* AA* = A*, (AA*)* = AA*, (A*A)* = A+ A. We also have that x = A+b is the shortest length least square solution to the problem Ax = b. (i) Show that if (A* A)-1 exists, then A+ = (A* A)-1A*. (ii) Let %3D 1 3 2 4 A = 3 5 Find the Moore-Penrose matrix inverse A+ of A.
Transcribed Image Text:Let A be an m × n matrix over C. The Moore-Penrose pseudo inverse matrix A+ is the unique n x m matrix which satisfies AA* A = A, A* AA* = A*, (AA*)* = AA*, (A*A)* = A+ A. We also have that x = A+b is the shortest length least square solution to the problem Ax = b. (i) Show that if (A* A)-1 exists, then A+ = (A* A)-1A*. (ii) Let %3D 1 3 2 4 A = 3 5 Find the Moore-Penrose matrix inverse A+ of A.
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