> In looking at some elementary linear algebra textbooks I noticed that > although they cover matrix inversion in the standard way, using row > reduction to go from [A | I] to [I | A^(-1)] , when they cover > inversion of block matrices they don't use the obvious analog. That > is, for example to find the inverse of say [[A,B],[0,C]] (with > appropriate conditions) they don't write this matrix augmented by the > block identity, [[I,0], [0,I]], and then do block row operations .. > for example multiplying the first row by A^(-1). > The usual approach these texbooks use is to compute the product of the > given matrix with some block matrix, say [[X,Y],[Z,W]], and equate the > result with the block identity and solve the resulting matrix > equations. > The first method I described seems simpler and has an obvious > connection with the method used to find inverses of numerical matrices > (Gauss-Jordan). It is true that the block row operations require a bit > more care, particularly with the order of multiplication, but is there > a more fundamental reason why these textbooks cover this topic the way > they do?
Perhaps one is just an algebraic rearangement of the other. ;-) The question then is which one provides more convenient formalae when you get to trying to implement the block result. Also notice that the block formulae tend to be applied to only a small numbers of blocks.