Symmetric equivalence of matrix polynomials and isolation of a common unital divisor in matrix polynomials (Q2754797)

The author considers factorizations of a symmetric polynomial matrix \(A(x)\in M_n(\mathbb C[x])\) of the form NEWLINE\[NEWLINE A(x)=B(x)C(x)B(x)^\nabla, \tag{1} NEWLINE\]NEWLINE where \(B(x)\) is invertible, and \(\nabla\) is an involution in \(\mathbb C[x]\). If (1) holds, \(A(x)\) and \(C(x)\) are called symmetrically equivalent. Conditions are found under which symmetric matrices are symmetrically equivalent to their Smith forms. Necessary and sufficient conditions are also found for the existence, for two matrix polynomials, of common unital divisors with given Smith forms.