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Topic: finding inverses of block matrices?
Replies: 1   Last Post: Nov 28, 2012 9:43 AM

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Posts: 38
Registered: 10/27/05
finding inverses of block matrices?
Posted: Nov 27, 2012 10:35 PM
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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
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?

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