I'm attacking to solve the graph isomorphism problem using the adjacent matrices. If two adjacent matrices of the input graphs $G_a$ and $G_b$ connect with a permutation matrix, then $G_a$ and $G_b$ are isomorphic.
Following statement is in the proof I’m working on. At first, I thought that it is trivial. However, I was worried that the following assumptions were correct.
Suppose the matrix $B$ consists $b_{i, j} = 1$ for $i \neq j$ and $b_{i, i} = 0$ (Then, $B$ is the adjacent matrix of the complete graph). When $P^t B P = B$, $P$ should be a permutation matrix.
I thought that since $P$ is symmetric, $P$ is orthogonal. $P$ is neither rotation nor inversion. Because the elements of $B$ are not changed. However, I wondered it is really correct.