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I understand how to perform SVD calculations, but am having difficulty with some of the theory. Specifically, if singular values are invariant when the original matrix is run through a similarity transformation. I know the eigenvalues are preserved, and have a feeling the singular values are not, but cannot find any resources on how to prove or disprove this.

Thank you in advance for any pointers or suggested resources for me to investigate.

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  • $\begingroup$ The singular values are unchanged if the original matrix is multiplied on the left by an orthogonal matrix and on the right by an orthogonal matrix. In general, though, the singular values change under non-orthogonal similarities. $\endgroup$ – kimchi lover Sep 28 '17 at 1:02
  • $\begingroup$ Is there something specific you could recommend for me to try and prove the "in-general" case? I am having a difficult time wrapping my brain around the conceptual of this, for some reason. $\endgroup$ – jthom Sep 28 '17 at 1:17
  • $\begingroup$ The $2\times2$ matrix of all 1s is similar to the matrix with entries $1$, $100$, $1/100$, $1$, and the singular values for these matrices are very different. $\endgroup$ – kimchi lover Sep 28 '17 at 1:23
  • $\begingroup$ Note that for all values of $t>0$, the matrices $$ \pmatrix{1 & t\\0&0} $$ are similar but have different singular values. $\endgroup$ – Omnomnomnom Sep 28 '17 at 1:40
  • $\begingroup$ Excellent, thanks to both of you. Both of those made total sense! I greatly appreciate the help. Linear algebra is the first bit of math that has truly confused me in a very long time. $\endgroup$ – jthom Sep 28 '17 at 1:47

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