Abstract
An orthogonality property common to a broad class of real symmetric matrix polynomials is developed generalizing earlier results concerning polynomials of second degree. This property is obtained with the help of canonical forms expressed in terms of a triple of real matrices (even though there may be complex spectrum) and it is used in the solution of an inverse spectral problem. The distribution of eigenvalues on the real line is discussed and earlier results for quadratic polynomials are generalized, in which the inertias of coefficient matrices are expressed in terms of the canonical forms.
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