Max and Min of matrix multiplication $ADA^{-1}$

Question

Suppose there's a matrix $A$ with no limits on its element values and a matrix $\Delta$ which is a diagonal matrix with elements that range between two values (say -1 and +1), is there a limit on the elements of the resulting matrix product $A \Delta A^{-1}$?

Answer 1

No. Counterexample: when $x\ne0$, we have $$ \frac12\pmatrix{1&1\\ -x&x}\pmatrix{1&0\\ 0&1/2}\pmatrix{1&-1/x\\ 1&1/x} =\frac14\pmatrix{3&-1/x\\ -x&3}, $$ which is unbounded when $x\to0$ or $x\to\infty$. You may even make all four entries unbounded by performing another similarity transform: $$ \pmatrix{1&1\\ 0&1}\left[\frac14\pmatrix{3&-1/x\\ -x&3}\right]\pmatrix{1&-1\\ 0&1} =\frac14\pmatrix{3-x&x-1/x\\ -x&3+x}. $$