What can you say about matrix $A\in M_n(\mathbb R)$ if you know that:
1) it is similar to its inverse matrix.
2) it is similar to its transpose matrix and similar to some diagonal matrices.
1) you write that $A=S^{-1}A^{-1}S$, I know that they have same eigenvalues, so $\det A\not =0$ $\operatorname{rank}A=n$, $\dim\ker(A)=0$, but can I say something more about this matrix, I saw that you can write like this $A=(AS)^{-1}S$, but I do not know is that help, can you help me?
2) $A=S^{-1}A^{T}S$ and $A=S^{-1}\Lambda S$, here I have no idea, do you have some idea? I read about similarity but I can not find so much about it, only that they have same rank, eigenvalues, characteristic polynomial.
Best Answer
1) On the one hand, $A$ and $A^{-1}$ have the same eigenvalues, as you observed. On the other hand, the eigenvalues of $A^{-1}$ are the reciprocals of those of $A$ (in general). So you can say quite a lot about the possible eigenvalues.
2) Prove that if $A$ is diagonalizable, then it satisfies this condition.