## Relation between singular values of matrices and their products

Hello everybody, 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 $\lambda_{XY^t}=f(\lambda_X , \lambda_Y)$?

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No. It matters how the singular vectors interact.

For example, let $X$ be the diagonal matrix with diagonal entries 1 and 2. Let $Y_1=X$, and let $Y_2$ be the diagonal matrix with diagonal entries 2 and 1. Then $XY_1$ and $XY_2$ have different singular values.

However, if you have estimates on the singular vectors, you may get estimates on the singular values of the product.

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Even though there is no functional relation between these singular values, there is a set of inequalities, called Horn inequalities, which completely describes possible singular values for $XY^t$ if the singular values of $X$ and $Y$ are fixed, see e.g. Fulton's survey ams.org/journals/bull/2000-37-03/… (the result is a corollary of Thompson's conjecture solved by Klyachko and work of Klyachko, Knutson and Tao on the eigenvalue problem). – Misha Jan 4 at 19:33