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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.

Show that if $A$ is an $n \times n$ positive definite matrix, then an orthogonal diagonalization $A=P D P^{T}$ is a singular value decomposition of $A .$

$A=P D P^{T}=U \Sigma V^{T}$ is a singular value decomposition

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

Oregon State University

Harvey Mudd College

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

01:25

$A$ is an $m \times n$ mat…

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05:05

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02:32

01:38

An $n \times n$ matrix $A$…

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A real $n \times n$ matrix…

05:57

Okay, so you miss program, we know that young gentlemen tricks, eh? Is positive. Definite. So were we record that, Uh, what's the positive definite means? So it means two things. The first wise a transport seacoast way. So ace a symmetric metrics and the the other day is a has I give values. Um, Lunda one up to London Onda uh, I didn't values. So Italian vectors Lee one up to the end. Such that a has low following the composition p the p chest post where p equals toward agonal magics with off this Eigen values on the diagonal entrance and the p it's a matrix with those again Lecter's. So now we're back to this, um, problem. This is the information we have. So we consider the metrics a trance post time Say we consider we can see that this matrix and we use this, um, again values and Magan vectors here. So, for example, H is post a times Any Eigen vectors off metrics A We wouldn't notice by be high. So I is from one to up to end. I'm sofrito this since we know that a is the symmetric matrix. So agents post time say you close to eight times eight times V I and we know that the I is an Agon director off a corresponding to wagon value and I So we have eight hands land I the eye sees Lambda is a constant So we have loved I times a v i again the eyes in agony value of a So we have London by square times the high So this tells us the ice Ah, I again vectors off hatreds post time, say, with Eigen value Eigen values Lum dies square eso basically tells us if we want to do the singular value decomposition for matrix A, um that that magics the should be You want each you out to the end, which is executive CMAs the metrics p we are using before. Okay, so now we are for the single ability competition. Rios. We also need to figure out what the metrics you. So by the definition, u equals two. Are you want u I equals to one over sig my eight tons v i but we know that the i r Eigen values argon vectors off magic, say, with Aiken Value Lunda. Hi. So it's one over. Sig my lambda High, But what's the lamb was listed my here. So we go back to this on this night, we know that the i r again their vectors off a transposed 10 say with Agon Burials Lambda Chi Square. That means the singular value Single Avengers off a are exactly land I So So we have a signal. My costume. Hamdi. So the result will be, um just sorry you should be London tons V I So the result of the V I That means the metrics you will be. You want the chew? Yeah, which is the same as p transposed. Ah, which is the p self? So sorry. It's not a piece of its p transpose. Okay, so the singular value decomposition uh, no is just people. So the single ability competition for a can be written as a close to your times? Um, sigma comes the transpose you will be p comes be times p transpose. Oh, it's Korean State to the regular regular decomposition for the metric say so we finish our prove here. So this decomposition off us off. Ah, positive. Definite Matrix A is the same as the singular value decomposition fla metrics

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