A Jordan Canonical Approach to an Indefinite Spectral Theorem

SMSU Event Center BC

One of the most profound and important theorems in Linear Algebra is the Spectral Theorem. This theorem is well-known for positive definite inner product spaces, but is less developed in indefinite inner product spaces.We aim to develop a generalization of an Indefinite Spectral Theorem by examining Jordan Form matrices and their associated metric. We start with an operator that is self-adjoint with respect to some inner product on a real vector space. We then complexify and extend the inner product to be Hermitian; from this, we determine the "best basis" for this matrix, forcing the metric to be a direct sum of Standard Involuntary Permutation matrices. The theorem we're after is simply a corollary of this result. This research has had applications in Differential Geometry, but would undoubtedly have further applications in Linear Algebra, as well.