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$A$ is an $m \times n$ matrix with a singular value decomposition $A=U \Sigma V^{T},$ where $U$ is an $m \times m$ orthogonal matrix, $\Sigma$ is an $m \times n$ 'diagonal" matrix with $r$ positive entries and no negative entries, and $V$ is an $n \times n$ orthogonal matrix. Justify each answer.

Suppose $A$ is square and invertible. Find a singular value decomposition of $A^{-1} .$

$A^{-1}=V \sum^{-1} U^{T}$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

Missouri State University

McMaster University

Lectures

01:32

In mathematics, the absolu…

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02:09

$A$ is an $m \times n$ mat…

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An $n \times n$ matrix $A$…

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A real $n \times n$ matrix…

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foreign problem, mate. Trouble 18. Um, we assume we have a convertible matrix. A where do universities, eh? You first. And we want to find a singular battled the composition of a chambers. So since we have Beatrice Kay which given and we know we can use the singular, better conversation to write A in such a form, it is you. You seem a and e transports. Excuse me. Now we consider the inverse. Okay, so we have you sigma and be transposed. Take the members now through the immersing size that we have transposed chambers and seek a metre seeking my members and you embers now that he's dead. Um, Matrix V is orthogonal, so the members will be the same as the transport. So that's the That's the same as taking the transpose off. Be twice. So he's just a baby and seeking my inverse and sees beach excuse also orthogonal. So he's just a u transpose now for sigalert Uber's now remember the embers of sigma starting the mystery signifies that diagonal matrix so uber is just to take the, um take one over. See my one on diagonal. It's the tickets taking the uber self Thea singular values. So this is our Sigma members. This all zeros, all dears. So here is our, ah, single about the composition off you.

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