Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation

We present the first systematic extension of the classical Hermite-Laguerre quadratic correspondence to the matrix-valued setting. Starting from a Hermite-type weight matrix W(x) = exp(-x^2) Z(x) with W(x) = W(-x), the change of variables y = x^2 produces two Laguerre-type weights with parameters alpha = -1/2 and alpha = 1/2, and relates the corresponding sequences of matrix-valued orthogonal polynomials through an explicit decomposition into even and odd subsequences. We prove that this transformation preserves differential operators and Darboux transformations, thereby establishing a direct structural link between the Hermite and Laguerre sides and providing new constructive tools for the matrix Bochner problem. Concrete families - including a new 3x3 example and an arbitrary-size family built from block-nilpotent matrices - illustrate the theory and supply fresh sources of matrix-valued orthogonal polynomials endowed with non-trivial differential algebras.