Suppose we are given a symmetric matrix $M$ of size $n\times n$ which I know to be diagonalized by the matrix $O$, i.e., $O^TMO$ is a diagonal matrix with eigenvalues of $M$.
Suppose I consider a submatrix $M'$ of $M$ (say the first $n/2$ rows and first $n/2$ columns of $M$) and I know that $M'$ is diagonalizable. My goal is to diagonalize $M'$. Is there any way I can diagonalize $M'$ by $O$? Or is there any generic way that I can diagonalize a submatrix (M') of a matrix (M) (which is known to be diagonalized by a matrix $O$)