Let \(\sigma \) be a Hermitian matrix measure supported on the unit circle. In this contribution, we study some algebraic and analytic properties of the orthogonal matrix polynomials associated with the Christoffel matrix transformation of \(\sigma \) defined by

$$\begin{aligned} d\sigma _{c_m}(z)=W_m(z)^Hd\sigma (z)W_m(z), \end{aligned}$$

where \(W_m(z)=\prod _{j=1}^m(z\mathbf{I} -A_j)\) and \(A_j\) is a square matrix for \(j=1,\ldots ,m.\) Moreover, we study the relative asymptotics of the associated orthogonal matrix polynomials when \(\sigma _{c_m}\) satisfies a matrix condition in the diagonal case. Some illustrative examples are considered.

中文翻译：

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关于单位圆上支持的矩阵测度的Christoffel变换

令\（\ sigma \）为单位圆上支持的Hermitian矩阵度量。在这项贡献中，我们研究了正交矩阵多项式的一些代数和解析性质，这些多项式与由（）定义的\（\ sigma \）的Christoffel矩阵变换有关

$$ \ begin {aligned} d \ sigma _ {c_m}（z）= W_m（z）^ Hd \ sigma（z）W_m（z），\ end {aligned} $$

其中\（W_m（z）= \ prod _ {j = 1} ^ m（z \ mathbf {I} -A_j）\）和\（A_j \）是\（j = 1，\ ldots的方阵， m。\）此外，我们研究当对角线情况下\（\ sigma _ {c_m} \）满足矩阵条件时，相关的正交矩阵多项式的相对渐近性。考虑一些说明性示例。