Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive semidefinite and Q is orthogonal.
Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive semidefinite and Q is orthogonal.
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
Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive semidefinite and Q is orthogonal.
Expert Solution
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Step 1
Given: Square matrix,
We need to show that every square matrix, can be factored as where -
Step 2
Let's write the single value decomposition of some matrix
Since is orthogonal matrix i.e. , we get-
Here, is orthogonally diagonalizable because is an orthogonal and is a diagonal matrix.
If a matrix is orthogonally diagonalizable, then it is symmetric.
is symmetric.
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