We study the interrelations between non-negative Hermitian \({q\times q}\) measures on the unit circle and non-negative Hermitian \({q\times q}\) measures on a symmetric interval \({[-2r,2r]}\) where \(r \in (0, \infty )\) . Since both types of non-negative Hermitian \({q\times q}\) measures are uniquely determined by their sequences of Fourier coefficients and power moments, respectively, we also express these interrelations in terms of moment sequences. Our approach significantly utilizes previous investigations by the authors on the inner structure of these sequences. Specifically, previous work by the authors employed Schur analysis methods to describe the intrinsic structure of \(\mathbb {T}\) -non-negative definite sequences of complex \({q\times q}\) matrices and \({[\alpha ,\beta ]}\) -non-negative definite sequences of complex \({q\times q}\) matrices. The primary aim of this paper is to uncover the relationships between corresponding parameters of sequences connected by a matrix version of the Szegő mapping, which was introduced by Szegő in the context of orthogonal polynomials. In essence, a complete concordance between both objects is achieved in the symmetric case.
