turkmath.org

A spectral theory for $n\times n$ selfadjoint matrix polynomials $L(\lambda)$ has been developed in recent decades. This will be surveyed and includes two lines of attack:
(a) using traditional Jordan structures and (b) using the eigenfunctions of the matrix-valued function $L(\lambda)$ .
Complete spectral data must include the idea of “sign characteristics” for real elementary divisors, and is included in the notion of a “real standard triple” - which provides a complete summary of spectral data.
A new orthogonality property will be presented - showing how half of the spectral data is related to the other half by a real orthogonal transformation.