We investigate the singular value decomposition (SVD) of a rectangular matrix $\boldsymbol{\mathit{A}}(z)$ of functions that are analytic on an annulus that includes at least the unit circle. Such matrices occur, e.g., as matrices of transfer functions representing broadband multiple-input multiple-output systems. Our analysis is based on findings for the analytic SVD applicable to continuous time systems, and on the analytic eigenvalue decomposition. Using these, we establish two potentially overlapping cases where analyticity of the SVD factors is denied. Firstly, from a structural point of view, multiplexed systems require oversampling by the multiplexing factor in order to admit an analytic solution. Secondly, from an algebraic perspective, we state under which condition spectral zeros of any singular value require additional oversampling by a factor of two if an analytic solution is to be found. In all other cases, an analytic matrix admits an analytic SVD, whereby the singular values are unique up to a permutation, and the left- and right-singular vectors are coupled through a joint ambiguity w.r.t. an arbitrary allpass function. We demonstrate how some state-of-the-art polynomial matrix decomposition algorithms approximate this solution, motivating the need for dedicated algorithms.

中文翻译：

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解析奇异值分解的性质和结构

我们研究矩形矩阵的奇异值分解（SVD）$\boldsymbol{\mathit{A}}(z)$在至少包括单位圆的环上解析的函数。这样的矩阵例如作为表示宽带多输入多输出系统的传递函数矩阵出现。我们的分析基于适用于连续时间系统的解析 SVD 的发现以及解析特征值分解。利用这些，我们建立了两个可能重叠的案例，其中 SVD 因子的分析性被否定。首先，从结构的角度来看，多路复用系统需要按多路复用因子进行过采样，以便获得解析解。其次，从代数的角度来看，我们指出在什么条件下，如果要找到解析解，任何奇异值的谱零点都需要两倍的额外过采样。在所有其他情况下，解析矩阵允许解析 SVD，由此奇异值在排列之前是唯一的，并且左奇异向量和右奇异向量通过任意全通函数的联合模糊度耦合。我们演示了一些最先进的多项式矩阵分解算法如何近似该解决方案，从而激发了对专用算法的需求。