Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $n\times 1$

share|improve this question
    
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
3  
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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.