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\iteman{io-port 05812422}
\itemau{Qiu, Yuyang; Zhang, Zhenyue; Wang, Anding}
\itemti{The least squares problem of the matrix equation $A_1X_1B^T_1+A_2X_2B^T_2=T$.}
\itemso{Appl. Math., Ser. B (Engl. Ed.) 24, No. 4, 451-461 (2009).}
\itemab
Summary: The matrix least squares (LS) problem $\min_X\|AXB^T-T\|_F$ is trivial and its solution can be simply formulated in terms of the generalized inverse of $A$ and $B$. Its generalized problem $\min_{X_1, X_2}\|A_1X_1B^T_1+A_2X_2B^T_2-T\|_F$ can also be regarded as the constrained LS problem $\min_{X=\text{diag}(X_1, X_2)}\|AXB^T-T\|_F$ with $A = [A_1, A_2]$ and $B = [B_1, B_2]$. The authors transform $T$ to $\widetilde{T}$ such that $\min_{X_1, X_2}\|A_1X_1B^T_1+A_2X_2B^T_2-T\|_F$ is equivalent to $\min_{X=\text{diag}(X_1, X_2)}\|AXB^T-\widetilde{T}\|_F$ whose solutions are included in the solution set of unconstrained problem $\min_X\|AXB^T-\widetilde{T}\|_F$. So, the general solutions of $\min_{X_1,X_2}\|A_1X_1B^T_1+A_2X_2B^T_2-T\|_F$ are reconstructed by selecting the parameter matrix in that of $\min_X\|AXB^T-\widetilde{T}\|_F$.
\itemrv{~}
\itemcc{}
\itemut{least squares problem; generalized inverse; solution set; general solutions; parameter matrix}
\itemli{doi:10.1007/s11766-009-2131-2}
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