Similar block diagonal Matrices

Question

I'm reading on the Jordan Normal Form of matrices theory and came upon the following result:

If $A_1$,$A_2$ are square matrices, then $A_2 \oplus A_1$ is similar to $A_1 \oplus A_2$.

I am trying to prove the above statement but I don't seem to be getting somewhere. Any suggestions/solution would be much appreciated.
Thank you.

Answer 1

Try to conjugate by a matrix of the form

$$\begin{pmatrix} A & I_n \\ I_m & B\end{pmatrix}$$

where $A$ is the $n\times m$ zero matrix, $B$ is the $m\times n$ zero matrix, and $I_n, I_m$ are the identity matrices of rank $n,m$, respectively.

As a start, show that the inverse of this matrix is its transpose. (This is a special permutation matrix, so conjugation by it permutes the basis vectors.)

Answer 2

Hint

The matrix $A_1\oplus A_2$ is a representation of a linear transformation $f:\Bbb R^n\to \Bbb R^n$ relatively to given basis $(e_1,\ldots,e_p,e_{p+1},\ldots,e_n)$ where $p$ is the size of $A_1$. Now, relatively to the basis $(e_{p+1},\ldots,e_n,e_1,\ldots,e_p)$, what's the matrix of $f$?