A[4,4] is divided into 4 blocks of an[2 2] then, Axx -> an(y,z)
Suppose A =(aij) nxn and A = ( Bpq ) where each B is a mxm matrix and m divides n. If n= m x N, then p and q run from 1 to N. Let i = m. I + s and j = m.J + t , then aij = (s, t) entry if s >0 or t >0 else s= m or t = m of BI+1, J+1 Block matrix.
I agree with Mittal, but given the hidden simplifying hypotheses that:
a)-m divides m (as also stated by Mittal)
b)-the matrixes A and B are square matrixes;
c)-the blocks B are not overlapping.
If one or all of the hypotheses falls, the thinks are more complicated, expecially near the corners of A.
firstly you must calculate the size of the main matrix A(r,c). Then calculate the number of the possible small matrices.Then by programming the solution is proposed by generating two nested loops, one for the external matrix (variable j) and the other for the internal matrices (variable k).
Let $M$ be a matrix partitioned into four blocks
\[
M = \left[\begin{array}{cc}
A &B\\
C &D
\end{array}\right]
\]
where the submatrix $A$ is assumed to be square and nonsingular. Brezinski asserted that, the \textcolor[rgb]{0.00,0.00,1.00}{\emph{Schur
complement}} of $A$ in $M$, denoted by $(M/A)$, is defined by
\begin{equation}\label{eq-Schur-1}
(M/A) = D-CA^{-1}B.
\end{equation}
When $M$ is square, matrices of this form are connected with block Gaussian decomposition, since we have
\begin{equation}\label{eq-Schur-2}
M = \left[\begin{array}{cc}
A &B\\
C &D
\end{array}\right]
= \left[\begin{array}{cc}
I &O\\
CA^{-1} &I
\end{array}\right]\left[\begin{array}{cc}
A &B\\
O &(M/A)
\end{array}\right].
\end{equation}
Moreover
\begin{equation}\label{eq-Schur-3}
|M|= |A||(M/A)|
\end{equation}
where $|\cdot|$ denotes determinant, an identity first proved by Schur.
See for example:
C. Brezinski, Other Manifestations of the Schur Complement, \emph{Linear Algebra and its Applications}, \textbf{111}, (1988), 231-247.
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