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Does the Cauchy's interlacing theorem hold for "singular values" of matrices too? I saw on this publication first Theorem that it does. It states that singular values of a matrix interlace the singular values of its principal sub-matrices. I would have thought given the original (celebrated) Cauchy's interlacing theorem that is on the "eigenvalues" of symmetric matrices and their sub-matrices, that to make interlacing statements about singular values we would need a restriction on positivity of the matrix. Is my intuition wrong?

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Your intuition is wrong; singular values are "nice" that way.

In particular: suppose that $A$ can be divided as $$ A = \pmatrix{A_0 & B\\C & D} $$ The interlacing property compares the singular values of $A$ to the singular values of $A_0$. That is, we are comparing the eigenvalues of $A^*A$ to the eigenvalues of $A_0^*A_0$. However, $A^*A$ has the structure $$ A^*A = \pmatrix{ A_0^*A + C^*C & A_0^*B + C^*D\\ B^*A_0 + D^*C & B^*B + D^*D} $$ Perhaps you can see how, at least in the case of $C = 0$, the Cauchy interlacing property applies directly. Remember that we can equivalently consider $AA^*$ and $AA^*$.

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