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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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Schur complement defines an accurate way to solve the linear equation involving super matrices. Any other improved version or algorithm will be very conducive. –  Abhinav Nov 2 '11 at 4:45
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. –  user13838 Nov 2 '11 at 11:11

1 Answer 1

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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So there is a requirement to analyze this system formally. of course we may stack the coeff matrix of n*n matrices into a single 2n*2n matrix. but we need to give a formal description of the algorithm to solve this. How may this be done? for example if i wish to extend the Cramer's rule to solve a simple linear system of equation to this block linear system, how may this be achieved? –  Abhinav Nov 3 '11 at 9:49
Good grief! Don't bother with Cramer! Use Gaussian elimination or something on your block linear system; that's all there is to it. –  J. M. Nov 3 '11 at 10:00

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