We propose and justify a matrix reduction method for calculating the optimal approximation of an observed matrix $A \in {\mathbb C}^{m \times n}$ by a sum $\sum_{i=1}^p \sum_{j=1}^q B_iX_{ij}C_j$ of matrix products where each $B_i \in {\mathbb C}^{m \times g_i}$ and $C_j \in {\mathbb C}^{h_j \times n}$ is known and where the unknown matrix kernels $X_{ij}$ are determined by minimizing the Frobenius norm of the error. The sum can be represented as a bounded linear mapping $BXC$ with unknown kernel $X$ from a prescribed subspace ${\mathcal T} \subseteq {\mathbb C}^n$ onto a prescribed subspace ${\mathcal S} \subseteq {\mathbb C}^m$ defined respectively by the collective domains and ranges of the given matrices $C_1,\ldots,C_q$ and $B_1,\ldots,B_p$. We show that the optimal kernel is $X = B^{\dag}AC^{\dag}$ and that the optimal approximation $BB^{\dag}AC^{\dag}C$ is the projection of the observed mapping $A$ onto a mapping from ${\mathcal T}$ to ${\mathcal S}$. If $A$ is large $B$ and $C$ may also be large and direct calculation of $B^{\dag}$ and $C^{\dag}$ becomes unwieldy and inefficient. { The proposed method avoids} this difficulty by reducing the solution process to finding the pseudo-inverses of a collection of much smaller matrices. This significantly reduces the computational burden.

翻译：本文提出并论证了一种矩阵降维方法，用于计算观测矩阵 $A \in {\mathbb C}^{m \times n}$ 通过矩阵乘积和 $\sum_{i=1}^p \sum_{j=1}^q B_iX_{ij}C_j$ 的最优逼近，其中每个 $B_i \in {\mathbb C}^{m \times g_i}$ 与 $C_j \in {\mathbb C}^{h_j \times n}$ 为已知矩阵，而未知的矩阵核 $X_{ij}$ 通过最小化误差的 Frobenius 范数确定。该求和可表示为有界线性映射 $BXC$，其未知核 $X$ 从给定子空间 ${\mathcal T} \subseteq {\mathbb C}^n$ 映射到给定子空间 ${\mathcal S} \subseteq {\mathbb C}^m$，这两个子空间分别由已知矩阵 $C_1,\ldots,C_q$ 与 $B_1,\ldots,B_p$ 的集体定义域和值域所确定。我们证明最优核为 $X = B^{\dag}AC^{\dag}$，且最优逼近 $BB^{\dag}AC^{\dag}C$ 是观测映射 $A$ 在从 ${\mathcal T}$ 到 ${\mathcal S}$ 的映射空间上的投影。当 $A$ 为大矩阵时，$B$ 和 $C$ 也可能规模较大，直接计算 $B^{\dag}$ 和 $C^{\dag}$ 将变得繁琐且低效。所提出的方法通过将求解过程转化为计算一系列规模小得多的矩阵的伪逆，避免了这一困难，从而显著降低了计算负担。