A product
A
=
F
1
…
F
N
A=F_1\,\ldots \,F_N
of invertible block-diagonal matrices will be banded with a banded inverse:
A
i
j
=
0
A_ij=0
and also
(
A
−
1
)
i
j
=
0
(A^{-1})_{ij}=0
for
|
i
−
j
|
>
w
|i-j|>w
. We establish this factorization with the number
N
N
controlled by the bandwidths
w
w
and not by the matrix size
n
.
n.
When
A
A
is an orthogonal matrix, or a permutation, or banded plus finite rank, the factors
F
i
F_i
have
w
=
1
w=1
and we find generators of that corresponding group. In the case of infinite matrices, the
A
=
L
P
U
A=LPU
factorization is now established but conjectures remain open.