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?
[Math] Interlacing Theorem on Singular Values
eigenvalues-eigenvectorslinear algebramatricessingular values
Best 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 $A_0$. However, the singular values of $A$ are equal to the non-negative positive eigenvalues of the matrix $$ M = \pmatrix{0 & A^*\\ A & 0} = \pmatrix{0 & 0 & A_0^* & C^*\\0 & 0 & B^* & D^*\\ A_0 & B & 0 & 0\\C & D & 0 & 0}. $$ By applying the interlacing inequality to this larger symmetric matrix and its principal submatrix $$ M_0 = \pmatrix{0 & A_0^*\\A_0 & 0}, $$ we end up with the interlacing inequality for singular values.