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I read this in my textbook but couldn't understand why this is true:

For a real positive semi-definite matrix A, the singular values are the same as the eigenvalues.

Could someone please explain this to me? I can prove it when A is symmetric but can't when A is just a square matrix. Thanks!

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    $\begingroup$ This isn't true. $\endgroup$ – Git Gud Dec 8 '14 at 21:23
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    $\begingroup$ I'm confused, what "general case" are you describing? In the definitions with which I am familiar, a nonnegative definite matrix is necessarily symmetric. Are you assuming that the matrix is nonnegative definite with respect to an inner product other than the standard one? If so, then the standard singular values will not coincide with the eigenvalues; instead the singular values for the other inner product will coincide with the eigenvalues. $\endgroup$ – Ian Dec 8 '14 at 21:25
  • $\begingroup$ Sorry, forgot to mention that A is a square matrix $\endgroup$ – mysteryduck Dec 8 '14 at 21:35
  • $\begingroup$ Relationship between eigenvalues and singular values math.stackexchange.com/questions/1276283/… $\endgroup$ – dantopa Mar 26 '17 at 21:19
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A positive semi-definite matrix must be symmetric. See the wikipedia page on definition of positive semi-definite matrices

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    $\begingroup$ It is possible to define positive (semi-)definite matrices (in the real case) which need not be symmetric. The Question seems to make the distinction clearly enough. Since the Question is a year old at this point, please make an effort to address what was asked (rather than assert the Question does not make sense). $\endgroup$ – hardmath Oct 18 '16 at 22:50

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