Subgradient of function $f_p(\mathbf{A})$ that has as output the $p^{th}$ largest singular value

Question

Suppose I have a matrix $\mathbf{A} \in \mathbb{R}^{n \times n}$. Let $\{\sigma_1,\sigma_2,.....,\sigma_n \}$ be its $n$ singular values. Calculating a sub-gradient of its operator norm (largest singular value) is a well known result. Are more general results available i.e can we find the sub-gradient of the function ( of the matrix $A$) that produces as output the second largest singular value, third largest singular value and generally p-th largest singular value ($p=1,2,...,n$). Is it a well posed question at all.

Answer 1

As an application of the Min-Max Theorem You get $$\sigma_{j+1}=\min_{x_1,..,x_j\in\mathbb{R}^n}\quad\max_{x\in\mathbb{R}^n||x||=1\\(x,x_1)=...=(x,x_j)=0}||Ax||.$$ As You see for $j=0$ you get $\sigma_1=|||A|||$ where $|||.|||$ is the operator norm. Of course the singular values are numbered in non-increasing order.Here $j=0,..,n-1$. The result can be generalized to ( possibly infinite-dimensional) Hilbert-spaces and compact operators.