Could anyone provide a reference to Proof of Weyl's theorem for singular values ?
If $X$ and $Y$ are $m\times n$ ($m\geq n$) and the singular values are ordered in the decreasing order, we have
$$σ_{i+j−1}^{(X+Y)} ≤ σ_i^{(X)}+σ_j^{(Y)} \text { for } 1≤i,j≤n,i+j−1≤n.$$
I have the following reference which proves it for eigen values for Hermitian matrices. But I am not sure how to extend it to singular values. https://www.math.drexel.edu/~foucart/TeachingFiles/F12/M504Lect5.pdf