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Find the SVD of $A=\left[\begin{array}{ccc}{3} & {2} & {2} \\ {2} & {3} & {-2}\end{array}\right]\left[\text { Hint: Work with } A^{T} .\right]$

$A=V \Sigma^{T} U^{T}=\left[\begin{array}{cc}{1 / \sqrt{2}} & {-1 / \sqrt{2}} \\ {1 / \sqrt{2}} & {1 / \sqrt{2}}\end{array}\right]\left[\begin{array}{ccc}{5} & {0} & {0} \\ {0} & {3} & {0}\end{array}\right]\left[\begin{array}{ccc}{1 / \sqrt{2}} & {1 / \sqrt{2}} & {0} \\ {-1 / \sqrt{18}} & {1 / \sqrt{18}} & {-4 / \sqrt{18}} \\ {-2 / 3} & {2 / 3} & {1 / 3}\end{array}\right]$

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 4

The Singular Value Decomposition

Introduction to Matrices

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In mathematics, the absolu…

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it's over this program. Um, instead of finding the singular value decomposition off metrics eight directly, we focus on the singular value decomposition off a transpose. The reason is the reason we make this choice is that eight times a transpose is a two by two matrix, but a trans post time Cesar three by three matrix. So we choose a smaller matrix. Um, you'll be easier of us. Okay, so eight times a chance pokes equals to, uh, a two by two matrix. And the characteristic equation for this are two by two matrix is 17 miners Lambda Square miners, 64 courses euro. So again, Marios I'm the one in close to 25 love that we close to nine. That means less singular value. Singler values off a chance posts are sick. My wine cost of five in a signatory course 29 i, Sigmund, because of three eso the metrics kept a sigma will be 5003 and 00 which has the same size as any transpose in the for the AG. You Mattis um London one number two. We need to figure out the Agon vector corresponding toe. Those argumentative So for Lambda when we have a matrix. Ah, minus 888 minus eight. So it shows everyone to be one over route off to one over Would have to, um, for London to withdraw within something. In this case, we have a matrix eight off for. Engines are right. So we chose the two would be minus one over it. Off to one over would have to. So the metrics in the single ability competition with the V equals to be one of the two in the four u one u two. When your three for the metrics You we have you. Why in Khost? One of a sigma one. A chance post. Uh, remember where you folks So agents post no to a times. Have you won? It gives us one of the root off to one over route off to zero for you to We have one of a sigma chul and transpose the two. It gives us minus one over route off 18 one over. Route off 18 minus former route of aiding. And that we need one more vector to, um generate the metrics. You here. So for now we have u equals to you. one and the YouTube. And for you three, we can just find out another also normal vector which know which perpendicular to both u one u 27 Just simply choose your three to be minus 2/3 to third and the 1/3 so you can check that you want times your three equals zero in the utensils. Three courses here and the the norm of your three Costa one. So once we make this choice, we have a single very decomposition for a transpose equals the U times. Sick Montanes being t that gives us a has, um, single Reddick single Avella decomposition v times sick imagines post times you transpose based on our information before we have a following. Before we, uh, our decomposition for a So this is a B and, uh, sick of my chance. Post will be 500 to your 30 and the U transpose will be. So you is this so just need to take the transpose for that. So this is the solution to this problem

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