Help in reducing given matrix to a particular form

Question

I am given a matrix, which has been partitioned as follows $$\textbf{K} = \begin{bmatrix}\textbf{K}_{mm} & \textbf{K}_{ms}\\\textbf{K}_{sm} & \textbf{K}_{ss}\end{bmatrix}$$ Here,
$K_{mm}$ has size $m * m$,
$K_{ms}$ has size $m*n$,
Total matrix dimensions are $(m+n)*(m+n)$

Now, I want to reduce it to the following form via a product of elementary matrices

$$\begin{bmatrix}\textbf{K}_{mm} & \textbf{K}_{ms}\\\textbf{K}_{sm} & \textbf{K}_{ss}\end{bmatrix} \rightarrow \begin{bmatrix}\textbf{A} & \textbf{0} \\ \textbf{B} & \textbf{I}\end{bmatrix}$$

The book I am referring to (on Guyan Condesnation), says this is a consequence of Gauss-Jordan elimination. I do not see the relation between this reduction and the row reduced echelon form, which is the GJE product. Can someone shed light on this ?

Answer 1

Hmm, let's see $$ \left( \begin{array}{c | c} I & 0 \\ \hline -A_{BL} A_{TL}^{-1} & I \end{array} \right)\left( \begin{array}{c | c} A_{TL} & A_{TR} \\ \hline A_{BL} & A_{BR} \end{array} \right) = \left( \begin{array}{c | c} A_{TL} & A_{TR} \\ \hline 0 & \widetilde A_{BR} \end{array} \right) $$ That is a block version of Gauss-Jordon step.

Then $$ \left( \begin{array}{c | c} I & -A_{TR} \widetilde A_{BR}^{-1} \\ \hline 0 & I \end{array} \right) \left( \begin{array}{c | c} A_{TL} & A_{TR} \\ \hline 0 & \widetilde A_{BR} \end{array} \right) = \left( \begin{array}{c | c} A_{TL} & 0 \\ \hline 0 & \widetilde A_{BR} \end{array} \right) $$ which is also a block version of a Gauss-Jordan step.

This is better than what you were asked to do. And, I think, uses nicer notations!

(TL = top-left, etc.)

(Check my algebra. Sometimes I make small, stupid mistakes. But you get the idea.)

Answer 2

After thinking about this problem for a while, this is what the question was meant to have you observe:

Gauss-Jordan usually progresses from the left-most column to the right-most column, one column at a time. But there is no law that says you have to start with the left-most column. You could instead compute a transform to introduce zeroes off the diagonal of the last column. Then the next-to-last column, and so forth. The problem is probably meant to make you think through that, and how that relates to standard Gauss-Jordan.

If that was the intent, then this is actually a really nice problem.