TITLE: Simultaneously diagonalization of two matrices.
QUESTION [7 upvotes]: Let $A$ be a real symmetric matrix and $B$ a real positive-definite matrix. Is it possible to simultaneously diagonalize of $A$ and $B$? Thank you very much.

REPLY [10 votes]: If you consider $A,B$ as quadratic forms, then, of course, they are simultaneously diagonalizable. That is, there is an invertible $S$ s.t. $S^TBS=I,S^TAS=D$ where $D$ is a diagonal matrix.
Proof: diagonalize $B$ and $B^{-1/2}AB^{-1/2}$. Since $B^{-1/2}AB^{-1/2}$ is symmetric, there is an orthogonal $O$ s.t. $O^TB^{-1/2}AB^{-1/2}O=D$ where $D$ is diagonal. Finally take $S=B^{-1/2}O$.