<s>
In	O
mathematics	O
,	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
kind	O
of	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
if	O
all	O
the	O
entries	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
if	O
all	O
the	O
entries	O
below	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
.	O
</s>
<s>
Because	O
matrix	O
equations	O
with	O
triangular	B-Algorithm
matrices	I-Algorithm
are	O
easier	O
to	O
solve	O
,	O
they	O
are	O
very	O
important	O
in	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
By	O
the	O
LU	O
decomposition	O
algorithm	O
,	O
an	O
invertible	O
matrix	O
may	O
be	O
written	O
as	O
the	O
product	O
of	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
L	O
and	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
U	O
if	O
and	O
only	O
if	O
all	O
its	O
leading	O
principal	O
minors	O
are	O
non-zero	O
.	O
</s>
<s>
is	O
called	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
or	O
right	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
lower	O
or	O
left	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
is	O
commonly	O
denoted	O
with	O
the	O
variable	O
L	O
,	O
and	O
an	O
upper	O
or	O
right	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
is	O
commonly	O
denoted	O
with	O
the	O
variable	O
U	O
or	O
R	O
.	O
</s>
<s>
A	O
matrix	O
that	O
is	O
both	O
upper	O
and	O
lower	B-Algorithm
triangular	I-Algorithm
is	O
diagonal	B-Algorithm
.	O
</s>
<s>
Matrices	O
that	O
are	O
similar	B-Algorithm
to	O
triangular	B-Algorithm
matrices	I-Algorithm
are	O
called	O
triangularisable	O
.	O
</s>
<s>
A	O
non-square	O
(	O
or	O
sometimes	O
any	O
)	O
matrix	O
with	O
zeros	O
above	O
(	O
below	O
)	O
the	O
diagonal	B-Algorithm
is	O
called	O
a	O
lower	O
(	O
upper	O
)	O
trapezoidal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
is	O
lower	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
A	O
matrix	O
equation	O
in	O
the	O
form	O
or	O
is	O
very	O
easy	O
to	O
solve	O
by	O
an	O
iterative	O
process	O
called	O
forward	O
substitution	O
for	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	O
and	O
analogously	O
back	O
substitution	O
for	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
process	O
is	O
so	O
called	O
because	O
for	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	O
,	O
one	O
first	O
computes	O
,	O
then	O
substitutes	O
that	O
forward	O
into	O
the	O
next	O
equation	O
to	O
solve	O
for	O
,	O
and	O
repeats	O
through	O
to	O
.	O
</s>
<s>
In	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
,	O
one	O
works	O
backwards	O
,	O
first	O
computing	O
,	O
then	O
substituting	O
that	O
back	O
into	O
the	O
previous	O
equation	O
to	O
solve	O
for	O
,	O
and	O
repeating	O
through	O
.	O
</s>
<s>
A	O
matrix	O
equation	O
with	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
U	O
can	O
be	O
solved	O
in	O
an	O
analogous	O
way	O
,	O
only	O
working	O
backwards	O
.	O
</s>
<s>
Forward	O
substitution	O
is	O
used	O
in	O
financial	O
bootstrapping	O
to	O
construct	O
a	O
yield	B-Algorithm
curve	I-Algorithm
.	O
</s>
<s>
The	O
transpose	O
of	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
is	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
and	O
vice	O
versa	O
.	O
</s>
<s>
A	O
matrix	O
which	O
is	O
both	O
symmetric	O
and	O
triangular	O
is	O
diagonal	B-Algorithm
.	O
</s>
<s>
In	O
a	O
similar	B-Algorithm
vein	O
,	O
a	O
matrix	O
which	O
is	O
both	O
normal	B-Algorithm
(	O
meaning	O
A*A	O
=	O
AA*	O
,	O
where	O
A*	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
)	O
and	O
triangular	O
is	O
also	O
diagonal	B-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
by	O
looking	O
at	O
the	O
diagonal	B-Algorithm
entries	O
of	O
A*A	O
and	O
AA*	O
.	O
</s>
<s>
The	O
determinant	O
and	O
permanent	O
of	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
equal	O
the	O
product	O
of	O
the	O
diagonal	B-Algorithm
entries	O
,	O
as	O
can	O
be	O
checked	O
by	O
direct	O
computation	O
.	O
</s>
<s>
In	O
fact	O
more	O
is	O
true	O
:	O
the	O
eigenvalues	O
of	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
are	O
exactly	O
its	O
diagonal	B-Algorithm
entries	O
.	O
</s>
<s>
Moreover	O
,	O
each	O
eigenvalue	O
occurs	O
exactly	O
k	O
times	O
on	O
the	O
diagonal	B-Algorithm
,	O
where	O
k	O
is	O
its	O
algebraic	O
multiplicity	O
,	O
that	O
is	O
,	O
its	O
multiplicity	O
as	O
a	O
root	O
of	O
the	O
characteristic	O
polynomial	O
of	O
A	O
.	O
</s>
<s>
that	O
is	O
,	O
the	O
unique	O
degree	O
n	O
polynomial	O
whose	O
roots	O
are	O
the	O
diagonal	B-Algorithm
entries	O
of	O
A	O
(	O
with	O
multiplicities	O
)	O
.	O
</s>
<s>
To	O
see	O
this	O
,	O
observe	O
that	O
is	O
also	O
triangular	O
and	O
hence	O
its	O
determinant	O
is	O
the	O
product	O
of	O
its	O
diagonal	B-Algorithm
entries	O
.	O
</s>
<s>
If	O
the	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
a	O
(	O
upper	O
or	O
lower	O
)	O
triangular	B-Algorithm
matrix	I-Algorithm
are	O
all	O
1	O
,	O
the	O
matrix	O
is	O
called	O
(	O
upper	O
or	O
lower	O
)	O
unitriangular	O
.	O
</s>
<s>
However	O
,	O
a	O
unit	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
not	O
the	O
same	O
as	O
the	O
unit	B-Algorithm
matrix	I-Algorithm
,	O
and	O
a	O
normed	O
triangular	B-Algorithm
matrix	I-Algorithm
has	O
nothing	O
to	O
do	O
with	O
the	O
notion	O
of	O
matrix	O
norm	O
.	O
</s>
<s>
If	O
all	O
of	O
the	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
a	O
(	O
upper	O
or	O
lower	O
)	O
triangular	B-Algorithm
matrix	I-Algorithm
are	O
also	O
0	O
,	O
the	O
matrix	O
is	O
called	O
strictly	O
(	O
upper	O
or	O
lower	O
)	O
triangular	O
.	O
</s>
<s>
All	O
finite	O
strictly	O
triangular	B-Algorithm
matrices	I-Algorithm
are	O
nilpotent	B-Algorithm
of	O
index	O
at	O
most	O
n	O
as	O
a	O
consequence	O
of	O
the	O
Cayley-Hamilton	O
theorem	O
.	O
</s>
<s>
An	O
atomic	O
(	O
upper	O
or	O
lower	O
)	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
form	O
of	O
unitriangular	O
matrix	O
,	O
where	O
all	O
of	O
the	O
off-diagonal	O
elements	O
are	O
zero	O
,	O
except	O
for	O
the	O
entries	O
in	O
a	O
single	O
column	O
.	O
</s>
<s>
Such	O
a	O
matrix	O
is	O
also	O
called	O
a	O
Frobenius	O
matrix	O
,	O
a	O
Gauss	B-Algorithm
matrix	I-Algorithm
,	O
or	O
a	O
Gauss	O
transformation	O
matrix	O
.	O
</s>
<s>
A	O
matrix	O
that	O
is	O
similar	B-Algorithm
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
referred	O
to	O
as	O
triangularizable	B-Algorithm
.	O
</s>
<s>
Abstractly	O
,	O
this	O
is	O
equivalent	O
to	O
stabilizing	O
a	O
flag	O
:	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
are	O
precisely	O
those	O
that	O
preserve	O
the	O
standard	O
flag	O
,	O
which	O
is	O
given	O
by	O
the	O
standard	O
ordered	O
basis	O
and	O
the	O
resulting	O
flag	O
All	O
flags	O
are	O
conjugate	O
(	O
as	O
the	O
general	O
linear	O
group	O
acts	O
transitively	O
on	O
bases	O
)	O
,	O
so	O
any	O
matrix	O
that	O
stabilises	O
a	O
flag	O
is	O
similar	B-Algorithm
to	O
one	O
that	O
stabilizes	O
the	O
standard	O
flag	O
.	O
</s>
<s>
Any	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
is	O
triangularizable	B-Algorithm
.	O
</s>
<s>
In	O
fact	O
,	O
a	O
matrix	O
A	O
over	O
a	O
field	O
containing	O
all	O
of	O
the	O
eigenvalues	O
of	O
A	O
(	O
for	O
example	O
,	O
any	O
matrix	O
over	O
an	O
algebraically	O
closed	O
field	O
)	O
is	O
similar	B-Algorithm
to	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
This	O
can	O
be	O
proven	O
by	O
using	O
induction	O
on	O
the	O
fact	O
that	O
A	O
has	O
an	O
eigenvector	O
,	O
by	O
taking	O
the	O
quotient	O
space	O
by	O
the	O
eigenvector	O
and	O
inducting	O
to	O
show	O
that	O
A	O
stabilizes	O
a	O
flag	O
,	O
and	O
is	O
thus	O
triangularizable	B-Algorithm
with	O
respect	O
to	O
a	O
basis	O
for	O
that	O
flag	O
.	O
</s>
<s>
A	O
more	O
precise	O
statement	O
is	O
given	O
by	O
the	O
Jordan	O
normal	B-Algorithm
form	O
theorem	O
,	O
which	O
states	O
that	O
in	O
this	O
situation	O
,	O
A	O
is	O
similar	B-Algorithm
to	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
of	O
a	O
very	O
particular	O
form	O
.	O
</s>
<s>
The	O
simpler	O
triangularization	O
result	O
is	O
often	O
sufficient	O
however	O
,	O
and	O
in	O
any	O
case	O
used	O
in	O
proving	O
the	O
Jordan	O
normal	B-Algorithm
form	O
theorem	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
complex	O
matrices	O
,	O
it	O
is	O
possible	O
to	O
say	O
more	O
about	O
triangularization	O
,	O
namely	O
,	O
that	O
any	O
square	B-Algorithm
matrix	I-Algorithm
A	O
has	O
a	O
Schur	O
decomposition	O
.	O
</s>
<s>
similar	B-Algorithm
,	O
using	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
as	O
change	O
of	O
basis	O
)	O
to	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
;	O
this	O
follows	O
by	O
taking	O
an	O
Hermitian	O
basis	O
for	O
the	O
flag	O
.	O
</s>
<s>
A	O
set	O
of	O
matrices	O
are	O
said	O
to	O
be	O
if	O
there	O
is	O
a	O
basis	O
under	O
which	O
they	O
are	O
all	O
upper	B-Algorithm
triangular	I-Algorithm
;	O
equivalently	O
,	O
if	O
they	O
are	O
upper	O
triangularizable	B-Algorithm
by	O
a	O
single	O
similarity	O
matrix	O
P	O
.	O
Such	O
a	O
set	O
of	O
matrices	O
is	O
more	O
easily	O
understood	O
by	O
considering	O
the	O
algebra	O
of	O
matrices	O
it	O
generates	O
,	O
namely	O
all	O
polynomials	O
in	O
the	O
denoted	O
Simultaneous	O
triangularizability	O
means	O
that	O
this	O
algebra	O
is	O
conjugate	O
into	O
the	O
Lie	O
subalgebra	O
of	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
,	O
and	O
is	O
equivalent	O
to	O
this	O
algebra	O
being	O
a	O
Lie	O
subalgebra	O
of	O
a	O
Borel	O
subalgebra	O
.	O
</s>
<s>
The	O
basic	O
result	O
is	O
that	O
(	O
over	O
an	O
algebraically	O
closed	O
field	O
)	O
,	O
the	O
commuting	O
matrices	O
or	O
more	O
generally	O
are	O
simultaneously	O
triangularizable	B-Algorithm
.	O
</s>
<s>
As	O
for	O
a	O
single	O
matrix	O
,	O
over	O
the	O
complex	O
numbers	O
these	O
can	O
be	O
triangularized	O
by	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
This	O
is	O
generalized	O
by	O
Lie	O
's	O
theorem	O
,	O
which	O
shows	O
that	O
any	O
representation	O
of	O
a	O
solvable	O
Lie	O
algebra	O
is	O
simultaneously	O
upper	O
triangularizable	B-Algorithm
,	O
the	O
case	O
of	O
commuting	O
matrices	O
being	O
the	O
abelian	O
Lie	O
algebra	O
case	O
,	O
abelian	O
being	O
a	O
fortiori	O
solvable	O
.	O
</s>
<s>
More	O
generally	O
and	O
precisely	O
,	O
a	O
set	O
of	O
matrices	O
is	O
simultaneously	O
triangularisable	O
if	O
and	O
only	O
if	O
the	O
matrix	O
is	O
nilpotent	B-Algorithm
for	O
all	O
polynomials	O
p	O
in	O
k	O
non-commuting	O
variables	O
,	O
where	O
is	O
the	O
commutator	O
;	O
for	O
commuting	O
the	O
commutator	O
vanishes	O
so	O
this	O
holds	O
.	O
</s>
<s>
One	O
direction	O
is	O
clear	O
:	O
if	O
the	O
matrices	O
are	O
simultaneously	O
triangularisable	O
,	O
then	O
is	O
strictly	O
upper	O
triangularizable	B-Algorithm
(	O
hence	O
nilpotent	B-Algorithm
)	O
,	O
which	O
is	O
preserved	O
by	O
multiplication	O
by	O
any	O
or	O
combination	O
thereof	O
–	O
it	O
will	O
still	O
have	O
0s	O
on	O
the	O
diagonal	B-Algorithm
in	O
the	O
triangularizing	O
basis	O
.	O
</s>
<s>
The	O
sum	O
of	O
two	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
is	O
upper	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
two	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
is	O
upper	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
The	O
inverse	O
of	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
,	O
if	O
it	O
exists	O
,	O
is	O
upper	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
The	O
product	O
of	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
and	O
a	O
scalar	O
is	O
upper	B-Algorithm
triangular	I-Algorithm
.	O
</s>
<s>
Together	O
these	O
facts	O
mean	O
that	O
the	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
form	O
a	O
subalgebra	O
of	O
the	O
associative	O
algebra	O
of	O
square	B-Algorithm
matrices	I-Algorithm
for	O
a	O
given	O
size	O
.	O
</s>
<s>
Additionally	O
,	O
this	O
also	O
shows	O
that	O
the	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
can	O
be	O
viewed	O
as	O
a	O
Lie	O
subalgebra	O
of	O
the	O
Lie	O
algebra	O
of	O
square	B-Algorithm
matrices	I-Algorithm
of	O
a	O
fixed	O
size	O
,	O
where	O
the	O
Lie	O
bracket	O
[	O
a	O
,	O
b ]	O
given	O
by	O
the	O
commutator	O
.	O
</s>
<s>
The	O
Lie	O
algebra	O
of	O
all	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
is	O
a	O
solvable	O
Lie	O
algebra	O
.	O
</s>
<s>
It	O
is	O
often	O
referred	O
to	O
as	O
a	O
Borel	O
subalgebra	O
of	O
the	O
Lie	O
algebra	O
of	O
all	O
square	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
All	O
these	O
results	O
hold	O
if	O
upper	B-Algorithm
triangular	I-Algorithm
is	O
replaced	O
by	O
lower	B-Algorithm
triangular	I-Algorithm
throughout	O
;	O
in	O
particular	O
the	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	O
also	O
form	O
a	O
Lie	O
algebra	O
.	O
</s>
<s>
However	O
,	O
operations	O
mixing	O
upper	O
and	O
lower	B-Algorithm
triangular	B-Algorithm
matrices	I-Algorithm
do	O
not	O
in	O
general	O
produce	O
triangular	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
sum	O
of	O
an	O
upper	O
and	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
can	O
be	O
any	O
matrix	O
;	O
the	O
product	O
of	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
with	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
is	O
not	O
necessarily	O
triangular	O
either	O
.	O
</s>
<s>
The	O
set	O
of	O
strictly	O
upper	O
(	O
or	O
lower	O
)	O
triangular	B-Algorithm
matrices	I-Algorithm
forms	O
a	O
nilpotent	B-Algorithm
Lie	O
algebra	O
,	O
denoted	O
This	O
algebra	O
is	O
the	O
derived	O
Lie	O
algebra	O
of	O
,	O
the	O
Lie	O
algebra	O
of	O
all	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
;	O
in	O
symbols	O
,	O
In	O
addition	O
,	O
is	O
the	O
Lie	O
algebra	O
of	O
the	O
Lie	O
group	O
of	O
unitriangular	O
matrices	O
.	O
</s>
<s>
In	O
fact	O
,	O
by	O
Engel	O
's	O
theorem	O
,	O
any	O
finite-dimensional	O
nilpotent	B-Algorithm
Lie	O
algebra	O
is	O
conjugate	O
to	O
a	O
subalgebra	O
of	O
the	O
strictly	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
,	O
that	O
is	O
to	O
say	O
,	O
a	O
finite-dimensional	O
nilpotent	B-Algorithm
Lie	O
algebra	O
is	O
simultaneously	O
strictly	O
upper	O
triangularizable	B-Algorithm
.	O
</s>
<s>
Algebras	O
of	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
have	O
a	O
natural	O
generalization	O
in	O
functional	B-Application
analysis	I-Application
which	O
yields	O
nest	B-Algorithm
algebras	I-Algorithm
on	O
Hilbert	O
spaces	O
.	O
</s>
<s>
The	O
set	O
of	O
invertible	O
triangular	B-Algorithm
matrices	I-Algorithm
of	O
a	O
given	O
kind	O
(	O
upper	O
or	O
lower	O
)	O
forms	O
a	O
group	O
,	O
indeed	O
a	O
Lie	O
group	O
,	O
which	O
is	O
a	O
subgroup	O
of	O
the	O
general	O
linear	O
group	O
of	O
all	O
invertible	O
matrices	O
.	O
</s>
<s>
A	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
invertible	O
precisely	O
when	O
its	O
diagonal	B-Algorithm
entries	O
are	O
invertible	O
(	O
non-zero	O
)	O
.	O
</s>
<s>
Over	O
the	O
real	O
numbers	O
,	O
this	O
group	O
is	O
disconnected	O
,	O
having	O
components	O
accordingly	O
as	O
each	O
diagonal	B-Algorithm
entry	O
is	O
positive	O
or	O
negative	O
.	O
</s>
<s>
The	O
identity	O
component	O
is	O
invertible	O
triangular	B-Algorithm
matrices	I-Algorithm
with	O
positive	O
entries	O
on	O
the	O
diagonal	B-Algorithm
,	O
and	O
the	O
group	O
of	O
all	O
invertible	O
triangular	B-Algorithm
matrices	I-Algorithm
is	O
a	O
semidirect	O
product	O
of	O
this	O
group	O
and	O
the	O
group	O
of	O
diagonal	B-Algorithm
matrices	I-Algorithm
with	O
on	O
the	O
diagonal	B-Algorithm
,	O
corresponding	O
to	O
the	O
components	O
.	O
</s>
<s>
The	O
Lie	O
algebra	O
of	O
the	O
Lie	O
group	O
of	O
invertible	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
is	O
the	O
set	O
of	O
all	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
,	O
not	O
necessarily	O
invertible	O
,	O
and	O
is	O
a	O
solvable	O
Lie	O
algebra	O
.	O
</s>
<s>
The	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
are	O
precisely	O
those	O
that	O
stabilize	O
the	O
standard	O
flag	O
.	O
</s>
<s>
The	O
group	O
of	O
invertible	O
lower	B-Algorithm
triangular	I-Algorithm
matrices	O
is	O
such	O
a	O
subgroup	O
,	O
since	O
it	O
is	O
the	O
stabilizer	O
of	O
the	O
standard	O
flag	O
associated	O
to	O
the	O
standard	O
basis	O
in	O
reverse	O
order	O
.	O
</s>
<s>
The	O
stabilizer	O
of	O
a	O
partial	O
flag	O
obtained	O
by	O
forgetting	O
some	O
parts	O
of	O
the	O
standard	O
flag	O
can	O
be	O
described	O
as	O
a	O
set	O
of	O
block	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
(	O
but	O
its	O
elements	O
are	O
not	O
all	O
triangular	B-Algorithm
matrices	I-Algorithm
)	O
.	O
</s>
