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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 *Gl*_{r} contained in
:

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

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

Here
denotes the normalizer in *Gl*_{r}(*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*^{-1}*VS*'; 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 *Q*_{J} 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
,

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