Is there any explicit relation between the singular values $\lambda_X$ and $\lambda_Y$ of two same size matrices $X$ and $Y$, respectively, and the singular values $\lambda_{XY^t}$ of the matrix $XY^t$? Otherwise said, is there a function $f$ such that $\lambda_{XY^t}=f(\lambda_X , \lambda_Y)$?
Thank you Riadh
Denote the $i$-th largest singular value of a matrix $A$ by $\sigma_i(A)$. Then for any two complex $m\times n$ matrices, we have the following inequality: $$ \sigma_{i+j-1}(XY^\ast)\le\sigma_i(X)\sigma_j(Y) \ \text{ when }\ i+j\le\min(m,n)+1. $$ See p.423 of Horn and Johnson's Matrix Analysis, Cambridge University Press.
As matrices are not completely characterized by their singular values, it is hard to believe that the singular values of $XY^T$ in general can be expressed as a function of the singular values of $X$ and $Y$. For example, consider $X=\operatorname{diag}(1,0),\,Y_1=X$ and $Y_2=\operatorname{diag}(0,1)$. Then $Y_1$ and $Y_2$ have identical singular values but $XY_1^T$ and $XY_2^T$ do not.
