New Proofs and Extensions of Sylvester’s and Johnson’s Inertia Theorems to Non-hermitian Matrices

We present a new proof and extension of the classical Sylvester Inertia Theorem to a pair of non-Hermitian matrices which satisfies the property that any real linear combination of the pair has only real eigenvalues. In the proof, we embed the given problem in a one-parameter family of related problems and examine the eigencurves of the family. The proof requires only elementary matrix theory and the Intermediate Value Theorem. The same technique is then used to extend Johnson’s extension of Sylvester’s Theorem on possible values of the inertia of a product of two matrices.