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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.

Using the notation of Exercise $23,$ show that $A^{T} \mathbf{u}_{j}=\sigma_{j} \mathbf{v}_{j}$ for $1 \leq j \leq r=\operatorname{rank} A$

$A^{T} \mathbf{u}_{j}=\sigma_{j} \mathbf{v}_{j} \quad(1 \leq j \leq r)$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

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Okay, So for this program, we already know that has a single A value decomposition. New tabs, signal tons, veterans post. So from this, um, decomposition we know that a transpose equals toe your time sigma terms of each transpose there's another transposed outside in the by the law of the transpose off majors This we have the times sigma chest posts, times you transpose. So it's his you he's in also orthogonal matrix That means you transpose equals Do you immerse so, agent supposed to be closed toe v times sick much as post times you immerse that gives us a chance Post tamps you you close to three times Sigman transpose so we know that. But sigma he caused to signal one up to a Sigmar. Ah, so for the rest part, with the zero also signatures pose will be the same as Sigma. Ah, but it was different size which doesn't make because we only care about the ring. So the rank off sigma t is obvious are which, of course, to the rink off the angina matrix A Also, if we ride out this multiplication explicitly, we have h is post times. Um you want you chew up to U N equals two The one the two up to VN Terms Sigma one up to Sigmar 000 eso we have a transpose times you Jay because toe VJ Times stigma J for and e j Jim one in the arm. This proves the statement in, uh for Britain.

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