Consider the matrix $A = 0$ that is diagonalized by the matrix
$$S = \begin{bmatrix} 5 & 2 \\ 2 & 1 \end{bmatrix}.$$
What is the diagonal matrix?
I'm confused because I thought you could just use the definition $S^{-1} A S = B$, in order to find the diagonal matrix $B$, and I ended up just getting the zero matrix, but I guess this is wrong. Any hints on what I'm doing wrong?
If $A$ is the $n\times n$ zero matrix $0_n$, it is already diagonal (non-diagonal elements all zero). Any invertible matrix of the appropriate size diagonalizes it, as well.
This is all trivial because $S^{-1}0_nS =0_n$, right?