The authors start by reviewing the definitions and details concerning sequences of matrix polynomials \(\{ P_{n}\left( x; \alpha \right) \}_{n \geq 0}\) orthonormal with respect to the following inner product (defined in the linear space of matrix polynomials \(\mathbb{C}^{\left(N,N \right)}[x]\)) \[ \langle P_{n}\left( \alpha \right) ,P_{m} \left( \alpha \right)\rangle_{L} = \int_{\Omega} P_{n}\left( x; \alpha \right) \, d\alpha(x) \, P_{m}\left( x; \alpha \right)^{\ast} =\delta_{n,m} I_{N}\quad n,m=0,1,2,\ldots, \] associated with an \(N \times N\) positive definite matrix \(\alpha(x)\) of measures supported on an infinite subset \(\Omega\) of the real line. In particular, such a sequence \(\{ P_{n}\left( x; \alpha \right) \}_{n \geq 0}\) fulfils the following recurrence relation \[ x P_{n}\left( x; \alpha \right) = A_{n+1}\left( \alpha \right) P_{n+1}\left( x; \alpha \right) + B_{n}\left(\alpha \right)P_{n}\left( x; \alpha \right)+A_{n}^{\ast} \left( \alpha \right)P_{n-1}\left( x; \alpha \right),\; n \geq 0, \] where \(B_{n}= \langle x P_{n}, P_{n} \rangle_{L} = \langle P_{n}, x P_{n} \rangle_{L}=B_{n}^{\ast}, \; n \geq 0,\) and \(A_{n}= \langle x P_{n-1}, P_{n} \rangle_{L},\; n \geq 1.\)
This contribution extends the results previously presented in [the first author, Integral Transforms Spec. Funct. 18, No. 1–2, 39–57 (2007; Zbl 1116.33013); the authors with M. A. Piñar, J. Approx. Theory 111, No. 1, 1–30 (2001; Zbl 1005.42014)] to the situations where:
\(\circ\;\) \( \lim\limits_{ n \rightarrow \infty} A_{n} \left( \alpha \right) = A\) (possibly singular) and \( \lim\limits_{ n \rightarrow \infty} B_{n} \left( \alpha \right) =B \);
\(\circ\;\) there is a sequence \(\{C_{n}\}_{n \geq 0}\) of positive definite matrices fulfilling \[ \lim_{ n \rightarrow \infty} C_{n}^{-1/2} A_{n} C_{n}^{-1/2} = A,\; \lim_{ n \rightarrow \infty} C_{n}^{-1/2} B_{n} C_{n}^{-1/2} = B \text{ and } \lim_{ n \rightarrow \infty} C_{n}^{-1/2} C_{n-1}^{1/2} = I_{N}. \] The relative asymptotics of orthonormal matrix polynomials regarding a varying matrix of measures are also studied.