# Interlacing Theorem on Singular Values

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?

## 1 Answer

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 $$AA^*$$.