Linear Algebra - Singular Value Decomposition Problem

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  • #1
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Homework Statement



Find the SVD of

equation 1.PNG


Homework Equations




The Attempt at a Solution


I'm stuck
equation 2.PNG

equation 3.PNG

equation 4.PNG


My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5 to square root 5.

General Question - When finding the singular value...
(sigma_1)^2 = constant, why do we only consider the positive root
sigma_1 = sqrt(constant)
because the solution to the problem is
sigma_1 = +/- sqrt(constant)

Thanks for any help you can provide me.
 

Answers and Replies

  • #2
vela
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My question is why in the solution it originally finds u_2=[1/5,-2/5]' but then says u_2=[1/sqrt(5),-2/sqrt(5)]'. I don't see what math was done in the solution to change the denominator from 5 to square root 5.
You're looking for an orthonormal basis.

General Question - When finding the singular value...
(sigma_1)^2 = constant, why do we only consider the positive root
sigma_1 = sqrt(constant)
because the solution to the problem is
sigma_1 = +/- sqrt(constant)

Thanks for any help you can provide me.
What's the definition you're using of a singular value?
 
  • #3
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Sorry I don't understand. It has to be orthornormal to u_1 so taking the dot product with u_1 and u_2 has to be zero and that's were it comes from?

I'm talking about just when it finds sigma_1 and sigma_2 why don't we consider the negative square root into our calculation when we find SVD? Like when we form the sigma matrix it's the singular values in a diagonal matrix, so I just don't understand really why we don't consider the negative root.
 
  • #4
vela
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Sorry I don't understand. It has to be orthornormal to u_1 so taking the dot product with u_1 and u_2 has to be zero and that's were it comes from?
No.

I'm talking about just when it finds sigma_1 and sigma_2 why don't we consider the negative square root into our calculation when we find SVD? Like when we form the sigma matrix it's the singular values in a diagonal matrix, so I just don't understand really why we don't consider the negative root.
Look up the definition of a singular value..
 
  • #5
blue_leaf77
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why don't we consider the negative square root into our calculation when we find SVD?
Singular values of a ##A## is defined to be the positive square root of the eigenvalues of ##A^*A##.
Nevertheless, if you choose to use the negative ones, it will still give the same original matrix ##A##.
 

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