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.
Categories: math.NA cs.NA