OSR Journal of Student Research

Article Title

A Jordan Canonical Approach to an Indefinite Spectral Theorem


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.

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