Theorem 11.20.20B75DFE662A7

Proof. At each point, choose an $h_ i$-orthonormal eigenbasis of $K_ i$. The resulting product basis is $h$-orthonormal, and $\widehat K_ i$ has the positive eigenvalues of $K_ i$, repeated according to the dimensions of the other tensor factors. Hence $h(\widehat K_ i v,v)>0$ for every nonzero $v$, whether or not $v$ is decomposable.

By lemma 11.20.1, each diagonal block of $\alpha _{h,v}$ has positive trace and therefore a positive eigenvalue, so the block diagonal form has at least $k$ positive eigenvalues. Since $k\leq n$, theorem 11.18.2 with $s=k$ gives the assertion. $\square$