Give a $n\times n$ matrix $H$. Suppose there exist a matrix $n\times n$ $A$ such that $A^{T}HA = D$ where $D$ is some diagonal matrix. Can we conclude:
1) $H$ is diagonalizable, i.e. there exists $B$ such that $B^{-1}HB = U$, where $U$ is diagonal 2) $A = B$ and $D = U$ (up to some row re-ordering as necessary)