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2. Block type Bruhat decomposition

For the convenience of the reader, we include some elementary, well-known facts from linear algebra which we shall need.

(2.1) Fix an integer and let

After a permutation of rows (or columns), any regular matrix will belong to .
Consider the following subgroups of Glr contained in :

where I always denotes the unit matrix of the corresponding size.

(2.2) The following lema will be verified by direct computation:

Lemma 1
Using the notation above,
(i1) ,
(i2) ,
(i3) .

Here denotes the normalizer in Glr(R).
(i1) Just check the following identities:

(i2) From the identity

assuming that the first matrix is in , we obtain T'=-S-1 T W' and V'=-W-1VS'; hence, SS'-TW-1 VS'=I. Thus, S' is invertible and is its inverse. Similarly, is the inverse of W' .
(i3) Using the notation of (i2) we obtain

(2.3) Fix a k-subset J of row indices . Let QJ be the submatrix of Q formed by those rows of Q whose index belongs to J. Then is invariant under equivalence of matrices. After a certain permutation of rows of Q, it is enough to consider for simplicity only the special cases ,

Next: 3. Row-minimal matrices Up: Splitting algorithm for vector Previous: 1. Introduction and notation
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