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How to solve a linear equation involving super matrices: $AX=B$. Is there any pre-existing algorithm?

where $A,X,B$ are super matrices i.e. matrices with elements which are simple matrices. in simplest case say $A=[ a(i,j)| i,j={1,2}];\ X=[ x(i)| i={1,2} ];\ \ B= [b(i) |i={1,2}] $ $a(i,j), x(i), b(i)$ are simple matrices of order $n\times n, n\times 1$ and $n\times 1$.

When $n=1$, it degenerates to simple system of linear equation.

Example: Solve for matrices $x,y$: $Ax+By=Q, Cx+Dy=T$

$A,B,C,D$ are matrices of order $n\times n$; $x,y,Q,T$ are of order $n\times 1$

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    Please, don't introduce this jargon. It is simply a linear matrix equation. It is usually referred to a block matrix if the entries are themselves matrices. Super and simple matrices are not contributing anything to the nature of the problem.2011-11-02

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It is perfectly fine to treat block matrices as your usual garden-variety matrices. To wit, the system of equations

$\begin{align*}\mathbf A\mathbf x+\mathbf B\mathbf y&=\mathbf f\\\mathbf C\mathbf x+\mathbf D\mathbf y&=\mathbf g\end{align*}$

and the block linear system

$\begin{pmatrix}\mathbf A&\mathbf B\\\mathbf C&\mathbf D\end{pmatrix}\cdot\begin{pmatrix}\mathbf x\\\mathbf y\end{pmatrix}=\begin{pmatrix}\mathbf f\\\mathbf g\end{pmatrix}$

are equivalent.

Thus, to solve for $\mathbf x$ and $\mathbf y$ in the system given above, just join/stack your four $n\times n$ matrices and two $n$-vectors appropriately and use your favorite linear equation solving algorithm to obtain a $2n$-vector that you can split in half.

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    Good grief! **Don't bother with Cramer!** Use Gaussian elimination or something on your block linear system; that's all there is to it.2011-11-03