I'm attempting to find a 4x4 matrix, P, that will convert my matrices,
$A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$
and,
$B = \begin{bmatrix}1&-1&1&1\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}$
into block diagonal matrices. I can see that A is already a block diagonal matrix (with a 2x2 block and then two 1x1 blocks).
Does this mean I only need to find a matrix P that converts B into block diagonal form and I am done or do I have to solve the problem simultaneously for both matrices?
I have found that I can fully diagonalise A by,
$S = \begin{bmatrix}0&0&-1&1\\0&0&2&0\\0&1&0&0\\1&0&0&0\end{bmatrix}$
and I have also proved that B is not diagonalisable.
Any help in how to block diagonalise B alone or even better how to block diagonalise A and B simultaneously would be much appreciated.
Thanks