Bounds for the squared deviations of the eigenvalues of two matrices in terms of any norm, preferably Frobenius norm or Max norm

Question

Let $A$ and $B$ be two real symmetric matrices of order $n$, with ordered eigenvalues $\lambda_i$ and $\mu_i$ for $i=1,2,\cdots,n$, respectively in decreasing order. How to find the bounds for the sum $S=\sum\limits_{i=1}^{n}(\lambda_i-\mu_i)^2$ in terms of $n$? In my problem we have both $A$ and $B$ having trace zero. Shall we find the bound with the help of any matrix norm?

Answer 1

The Frobenius norm is useful to measure the RMS (root-mean-square) gain of the matrix, its average response along given mutually orthogonal directions in space. Clearly, this approach does not capture well the variance of the error, only the average effect of noise.