Comapring singular values of two matrices

Question

If I have two matrices of type $n\times m,$ is there a way to compare their singular values? For example, if $A$ has singular values $\lambda_1,\cdots,\lambda_n,$ $\lambda_1\geqslant\lambda_2\geqslant\cdots\geqslant\lambda_n\geqslant0,$ and $B$ has singular values $\sigma_1,\ldots,\sigma_n,$ $\sigma_1\geqslant\sigma_2\geqslant\cdots\geqslant\sigma_n\geqslant0,$ is there a way to compare $\lambda_1+\cdots+\lambda_n$ and $\sigma_1+\cdots+\sigma_n$ or $\lambda_1+\cdots+\lambda_n$ and $\sigma_1^2+\cdots+\sigma_n^2?$