This post https://mathoverflow.net/questions/49679/a-matrix-similarity-problem makes the claim that conjugating by an upper triangular matrix does not change the diagonal entries. But how do I prove this for all n x n matrices?
Also, what about the converse. Suppose A,B,C are invertible matrices and $ABA^{-1} = C$ where $B$ and $C$ have the same diagonal. Must it be the case that $A$ is triangular?
The linked Mathoverflow question doesn’t claim that conjugation of any matrix by an upper triangular matrix preserves the diagonal, and such interpretation would be, in fact, not true.
On the other hand, conjugation of the identity matrix by any invertible matrix gives also the identity matrix. Not to say the zero matrix also gives zero matrices under any multiplications.
So, −1.
the considered result is true only if the first matrix is also triangular.
