Spectral norm of difference of two similar matrices

Question

Suppose we have two similar matrices $R$ and $L$.

1. Is there a bound on the spectral norm of $R-L$ using the singular values of $R$ and $L$?

Suppose $R^TR$ and $L^TL$ are similar matrices:

2. Can we find a bound on the spectral norm of $R-L$ using the biggest eigenvalue of $R^TR$?

3. What if $R$ and $L$ are random matrices whose ESD fits an MP distribution? Can one find a bound on the spectral norm of $R-L$ using the largest eigenvalue in the ESD of R and the size of R?

Answer 1

If $R$ and $L$ are merely similar, $\|R-L\|_2$ is not bounded above. Consider $R=\pmatrix{1&0\\ 0&-1}$ and $L=\pmatrix{1&t\\ 0&-1}$ for instance.

If $R^TR$ is similar to $L^TL$ and $L,R$ are real matrices, then $L$ and $R$ have the same singular values. It follows from singular value decomposition that $L=URV$ for some orthogonal matrices $U$ and $V$. Therefore $\|L-R\|\le\|L\|+\|R\|=2\|R\|$. Equality is attained when, for instance, $L=-R$.