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If $A$ is a $3 \times 3$ matrix with singular values $5$, $4$, and $2$, then there are $9$ distinct singular value decompositions of $A$. True or false?

Is there any method to solve this because I'm not sure how to approach this.

Rodrigo de Azevedo
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Tofuine
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    Sometimes it helps to get started by considering the simplest possible example, in this case a $3\times 3$ matrix with the designated singular values. If you plan to participate in Math.SE in an on-going fashion, you'll probably want to learn at least the rudiments of posting typeset mathematical expressions. – hardmath Jul 12 '20 at 15:42
  • Oh, I know a lot about latex, I just thought it would be ok to write numbers without putting them in latex, but ok! Also, I'll try to do it that way using an example and then get back to you. – Tofuine Jul 12 '20 at 15:45
  • Actually, how do I make a matrix with the singular values of $5$, $4$, and $2$ from scratch? – Tofuine Jul 12 '20 at 15:47
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    Try writing out the definition of SVD. See what happens if you permute the diagonal entries of singular values. Then count the number of distinct permutations – Dinesh Jul 12 '20 at 15:47
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    Just go through this link. http://rakaposhi.eas.asu.edu/s10-cse494-mailarchive/msg00030.html – AxyuS Jul 12 '20 at 15:54
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    Is $A$ real or complex? For example, $U \Sigma V^T = (-U) \Sigma (-V)^T$. – copper.hat Jul 12 '20 at 15:56
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    @Dr.Disambiguous To see how people typeset something, find an example on Math.SE and left click, select Show Math As->TeX Commands to see. – copper.hat Jul 12 '20 at 16:00
  • @AxyuS I looked at the link but I'm not sure what df, ff, so on are. – Tofuine Jul 12 '20 at 16:10
  • @copper.hat I know how to typeset things. Also, thanks for the tip! I'm assuming $A$ is real. – Tofuine Jul 12 '20 at 16:11
  • I have a question: Does the middle matrix $\Sigma$ need to be in the order of greatest to least singular values, or can it be in any order? – Tofuine Jul 12 '20 at 16:15
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    Somewhat related: https://math.stackexchange.com/q/1805191/339790 – Rodrigo de Azevedo Jul 12 '20 at 16:57
  • @Dr.Disambiguous You must be aware of this equation $A = U S V^{T}$. Those symbols are nothing but different names for the variable of equation written above.$ – AxyuS Jul 14 '20 at 18:02

1 Answers1

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Important rule:

  • An $n*n$ matrix with n distinct positive singular values has $2^{n}$ different singular value decompositions (svds).

  • An $n*n$ singular matrix (one without an inverse) with $n$ distinct singular values ($\sigma_{n}$ can be $0$) has $2^{n+1}$ distinct svds.

  • An $n*n$ matrix with a repeated singular value has $\infty$ svds.

This problem satisfies the first rule, therefore $A$ has $2^{3}=8$ distinct svds.

Answer: False

Tofuine
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