We present a generalization of the pseudoinverse operation to pairs of matrices, as opposed to single matrices alone. We note the fact that the Singular Value Decomposition can be used to compute the ordinary Moore-Penrose pseudoinverse. We present an analogue of the Singular Value Decomposition for pairs of matrices, which we show is inadequate for our purposes. We then present a more sophisticated analogue of the SVD which includes features of the Jordan Normal Form, which we show is adequate for our purposes. This analogue of the SVD, which we call the Jordan SVD, was already presented in a previous paper by us called ‘Matrix decompositions over the double numbers’. We adopt the idea presented in that same paper that a pair of matrices is actually a single matrix over the double-complex number system.

我们将伪逆运算推广到矩阵对，而不是单独的单个矩阵。我们注意到奇异值分解可用于计算普通的 Moore-Penrose 伪逆。我们提出了矩阵对的奇异值分解的类似物，我们证明它不足以满足我们的目的。然后，我们提出一个更复杂的 SVD 模拟，其中包括 Jordan 范式的特征，我们证明它足以满足我们的目的。这种 SVD 的类似物，我们称之为 Jordan SVD，已经在我们之前的一篇名为“双数矩阵分解”的论文中提出过。我们采用同一篇论文中提出的想法，即一对矩阵实际上是双复数系统上的单个矩阵。