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Find an SVD of each matrix [Hint: In Exercise 11, one choice for $U$ is $\left[\begin{array}{rrr}{-1 / 3} & {2 / 3} & {2 / 3} \\ {2 / 3} & {-1 / 3} & {2 / 3} \\ {2 / 3} & {2 / 3} & {-1 / 3}\end{array}\right]$ In Exercise $12,$ one column of $U$ can be $\left[\begin{array}{c}{1 / \sqrt{6}} \\ {-2 / \sqrt{6}} \\ {1 / \sqrt{6}}\end{array}\right].$]$\left[\begin{array}{rr}{2} & {-1} \\ {2} & {2}\end{array}\right]$

$A=U \Sigma V^{T}=\left[\begin{array}{cc}{1 / \sqrt{5}} & {2 / \sqrt{5}} \\ {2 / \sqrt{5}} & {-1 / \sqrt{5}}\end{array}\right]\left[\begin{array}{cc}{3} & {0} \\ {0} & {2}\end{array}\right]\left[\begin{array}{cc}{2 / \sqrt{5}} & {1 / \sqrt{5}} \\ {1 / \sqrt{5}} & {-2 / \sqrt{5}}\end{array}\right]$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

Oregon State University

McMaster University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:32

In mathematics, the absolu…

01:11

05:53

Find an SVD of each matrix…

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03:55

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03:32

02:14

Repeat Exercise 15 for the…

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Find each matrix product i…

00:45

Find the transpose of each…

Okay, So for this question, we can see the metrics. A trance post, Hans A, uh which is 8 to 5 to two by two matrix. And then you can see there the Agon value off this metric state. So the characteristic equation for this matrix would be the determinant off a trance post times a minor Islam die, which is, um, we because zero. So, this equation, if we ride it out, you will be eight mine Islam that times five miners, Lambda minus four equals zero. And this quadratic equation has two solutions. Loved the one he close to nine lamb that two equals two. For so the singular value Singular values off man checks a are Sigma y in close to a three Sigma Tau requested to in the face on this singular values in the single A value decomposition off a the metrics kept a signal because to 3002 Also, this matrix has the same size as the own Geno metrics. A. So let's we have. It is Eigen values the Agon vectors for a transpose times a were bit of following so we can see Lamda won a transpose times a miner's Lambda one times I because to minus one True and the true minus four. So based on these metrics, we have the one he caused to two of a root of five. And the one over Ruutel five, which is in or throw normal? Um, Victor in the phylum. That too. We do the same thing. So the metrics is for 2 to 1. According to this metrics, we can set up another vector B two, which is one of a root of five. And the minus 2/5. And since we already have, we want me to Also in the secret of allergy composition of Matrix A. The metrics v equals June on tour route of 51 of a route off five one over. Route off five and a two over route five, minus two over to fashion and the ones we have. This B one B two, we can figure out the matrix. You. So you want he cost one over Sigma 18 times the one which is one over route. Tough too. A little five and a two over. Ruto five. You too, equals to one of a stigma to a times B two So it's ah to over route of five and the one a minus one over route five. Okay, so once they figure out this you want you to So the metrics you is the ally off. This Teoh is two factors and that the singular value decomposition is just a equals two matrix You times, matrix sigma turns magics, we transpose which is just right. The result out again. This result is not unique because we can use different Eigen values we wanted to. For example, we can put a minus sign in front of this. But if they change this Aiken vector, we need to change the baby on you one. So it's the matrix you but the result of evidence here. So this is a singular value. Decomposition of what metrics.

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