<s>
In	O
mathematics	O
,	O
Birkhoff	B-Algorithm
factorization	I-Algorithm
or	O
Birkhoff	O
decomposition	O
,	O
introduced	O
by	O
,	O
is	O
the	O
factorization	O
of	O
an	O
invertible	O
matrix	O
M	O
with	O
coefficients	O
that	O
are	O
Laurent	O
polynomials	O
in	O
z	O
into	O
a	O
product	O
M	O
=	O
M+M0M−	O
,	O
where	O
M+	O
has	O
entries	O
that	O
are	O
polynomials	O
in	O
z	O
,	O
M0	O
is	O
diagonal	O
,	O
and	O
M−	O
has	O
entries	O
that	O
are	O
polynomials	O
in	O
z−1	O
.	O
</s>
<s>
There	O
are	O
several	O
variations	O
where	O
the	O
general	O
linear	B-Algorithm
group	I-Algorithm
is	O
replaced	O
by	O
some	O
other	O
reductive	O
algebraic	O
group	O
,	O
due	O
to	O
.	O
</s>
<s>
Birkhoff	B-Algorithm
factorization	I-Algorithm
implies	O
the	O
Birkhoff	O
–	O
Grothendieck	O
theorem	O
of	O
that	O
vector	O
bundles	O
over	O
the	O
projective	O
line	O
are	O
sums	O
of	O
line	O
bundles	O
.	O
</s>
<s>
Birkhoff	B-Algorithm
factorization	I-Algorithm
follows	O
from	O
the	O
Bruhat	O
decomposition	O
for	O
affine	O
Kac	O
–	O
Moody	O
groups	O
(	O
or	O
loop	O
groups	O
)	O
,	O
and	O
conversely	O
the	O
Bruhat	O
decomposition	O
for	O
the	O
affine	O
general	O
linear	B-Algorithm
group	I-Algorithm
follows	O
from	O
Birkhoff	B-Algorithm
factorization	I-Algorithm
together	O
with	O
the	O
Bruhat	O
decomposition	O
for	O
the	O
ordinary	O
general	O
linear	B-Algorithm
group	I-Algorithm
.	O
</s>
