Let $A$ and $B$ be similar real symmetric matrices with the same diagonal entries. Prove that $A=B$.
I not 100% sure that this is true but if it's true proves another problem that I'm trying to solve.
(Edit 1:) Let $A=[a_{ij} ]$ where $a_{ij}\in \{0, 1\}$ be a real matrix such that the diagonals of $AA^{t}$ and $A^{t}A$ are equal to $(k, k, \dots , k)$ and $AA^{t}=(k-c)I+cU$ where $U$ is the matrix where every entry is $1$ and $k\geq c$. Prove that $AA^{t}=A^{t}A$.