I am trying to relate eigenvalues with singular values. In particular,

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I am trying to relate eigenvalues with singular values. In particular, I'm trying to show that for any eigenvalue of A, it is within range of the singular values of A. In other words,

smallestSingularValue(A) <= |anyEigenValue(A)| <= largestSingularValue(A).

I've tried using Schur decomposition, and then permuting the matrix so that the eigenvalues are ordered like the singular values. But I can't determine their relationship. Any help would be appreciated.
 
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The singular values of A are equal to square roots of the eigenvalues of A^T.A.

Eigenvalues only exist for square matrices. Singular values exist for rectangular matrices as well as square ones.
 


well, this is true:
smallestSingularvalue(T)*|v| <= |Tv| <= largestsingularvalue(T)*|v|
try proving that
 

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