# Simultaneous diagonalization of a set of Hermitian matrices

In Horn's Matrix Analysis (Theorem 7.6.4), it is stated that

Let $A,B$ be two Hermitian matrices and suppose that there is a real linear combination of $A$ and $B$ that is positive definite. Then there exists a nonsingular matrix $C$ such that $C^{*}AC$ and $C^{*}BC$ are diagonal.

My question is whether there is an extension of this theorem in the literature that applies not just to two Hermitian matrices, but to an arbitrary finite set of Hermitian matrices.

Thanks!

There is no reason to believe that the statement can be generalised to the case of more than two matrices. In fact, if it can be generalised, then any two Hermitian matrices $A$ and $B$ can be simultaneously diagonalised by congruence because $I+0A+0B$ is positive definite and it is a real linear combination of $I,A,B$. But clearly, not every pair of Hermitian matrices can be simultaneously diagonalised by congruence. For a counterexample, consider $A=\pmatrix{1&0\\ 0&-1}$ and $B=\pmatrix{0&1\\ 1&0}$. It is not hard to show that if both $C^\ast AC$ and $C^\ast BC$ are diagonal, $C$ must be noninvertible.