<s>
In	O
linear	B-Language
algebra	I-Language
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
(	O
or	O
Gramian	B-Algorithm
matrix	I-Algorithm
,	O
Gramian	B-Algorithm
)	O
of	O
a	O
set	O
of	O
vectors	O
in	O
an	O
inner	O
product	O
space	O
is	O
the	O
Hermitian	B-Algorithm
matrix	I-Algorithm
of	O
inner	O
products	O
,	O
whose	O
entries	O
are	O
given	O
by	O
the	O
inner	O
product	O
.	O
</s>
<s>
If	O
the	O
vectors	O
are	O
the	O
columns	O
of	O
matrix	O
then	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
in	O
the	O
general	O
case	O
that	O
the	O
vector	O
coordinates	O
are	O
complex	O
numbers	O
,	O
which	O
simplifies	O
to	O
for	O
the	O
case	O
that	O
the	O
vector	O
coordinates	O
are	O
real	O
numbers	O
.	O
</s>
<s>
An	O
important	O
application	O
is	O
to	O
compute	O
linear	O
independence	O
:	O
a	O
set	O
of	O
vectors	O
are	O
linearly	O
independent	O
if	O
and	O
only	O
if	O
the	O
Gram	O
determinant	O
(	O
the	O
determinant	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
)	O
is	O
non-zero	O
.	O
</s>
<s>
For	O
finite-dimensional	O
real	O
vectors	O
in	O
with	O
the	O
usual	O
Euclidean	O
dot	O
product	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
,	O
where	O
is	O
a	O
matrix	O
whose	O
columns	O
are	O
the	O
vectors	O
and	O
is	O
its	O
transpose	O
whose	O
rows	O
are	O
the	O
vectors	O
.	O
</s>
<s>
For	O
complex	O
vectors	O
in	O
,	O
,	O
where	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
.	O
</s>
<s>
Given	O
square-integrable	B-Algorithm
functions	I-Algorithm
on	O
the	O
interval	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
:	O
</s>
<s>
For	O
any	O
bilinear	O
form	O
on	O
a	O
finite-dimensional	O
vector	O
space	O
over	O
any	O
field	O
we	O
can	O
define	O
a	O
Gram	B-Algorithm
matrix	I-Algorithm
attached	O
to	O
a	O
set	O
of	O
vectors	O
by	O
.	O
</s>
<s>
The	O
matrix	O
will	O
be	O
symmetric	B-Algorithm
if	O
the	O
bilinear	O
form	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
In	O
Riemannian	O
geometry	O
,	O
given	O
an	O
embedded	O
-dimensional	O
Riemannian	O
manifold	O
and	O
a	O
parametrization	O
for	O
the	O
volume	O
form	O
on	O
induced	O
by	O
the	O
embedding	O
may	O
be	O
computed	O
using	O
the	O
Gramian	B-Algorithm
of	O
the	O
coordinate	O
tangent	O
vectors	O
:	O
This	O
generalizes	O
the	O
classical	O
surface	O
integral	O
of	O
a	O
parametrized	O
surface	O
for	O
:	O
</s>
<s>
If	O
the	O
vectors	O
are	O
centered	O
random	O
variables	O
,	O
the	O
Gramian	B-Algorithm
is	O
approximately	O
proportional	O
to	O
the	O
covariance	O
matrix	O
,	O
with	O
the	O
scaling	O
determined	O
by	O
the	O
number	O
of	O
elements	O
in	O
the	O
vector	O
.	O
</s>
<s>
In	O
quantum	O
chemistry	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
a	O
set	O
of	O
basis	O
vectors	O
is	O
the	O
overlap	O
matrix	O
.	O
</s>
<s>
In	O
control	O
theory	O
(	O
or	O
more	O
generally	O
systems	O
theory	O
)	O
,	O
the	O
controllability	O
Gramian	B-Algorithm
and	O
observability	O
Gramian	B-Algorithm
determine	O
properties	O
of	O
a	O
linear	O
system	O
.	O
</s>
<s>
Gramian	B-Algorithm
matrices	O
arise	O
in	O
covariance	O
structure	O
model	O
fitting	O
(	O
see	O
e.g.	O
,	O
Jamshidian	O
and	O
Bentler	O
,	O
1993	O
,	O
Applied	O
Psychological	O
Measurement	O
,	O
Volume	O
18	O
,	O
pp.79	O
–	O
94	O
)	O
.	O
</s>
<s>
In	O
the	O
finite	B-Application
element	I-Application
method	I-Application
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
arises	O
from	O
approximating	O
a	O
function	O
from	O
a	O
finite	O
dimensional	O
space	O
;	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
entries	O
are	O
then	O
the	O
inner	O
products	O
of	O
the	O
basis	O
functions	O
of	O
the	O
finite	O
dimensional	O
subspace	O
.	O
</s>
<s>
Since	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
over	O
the	O
reals	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
it	O
is	O
diagonalizable	B-Algorithm
and	O
its	O
eigenvalues	O
are	O
non-negative	O
.	O
</s>
<s>
The	O
diagonalization	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
the	O
singular	O
value	O
decomposition	O
.	O
</s>
<s>
The	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
in	O
the	O
case	O
the	O
real	O
product	O
is	O
real-valued	O
;	O
it	O
is	O
Hermitian	B-Algorithm
in	O
the	O
general	O
,	O
complex	O
case	O
by	O
definition	O
of	O
an	O
inner	O
product	O
.	O
</s>
<s>
The	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
positive	B-Algorithm
semidefinite	I-Algorithm
,	O
and	O
every	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
is	O
the	O
Gramian	B-Algorithm
matrix	I-Algorithm
for	O
some	O
set	O
of	O
vectors	O
.	O
</s>
<s>
The	O
fact	O
that	O
the	O
Gramian	B-Algorithm
matrix	I-Algorithm
is	O
positive-semidefinite	O
can	O
be	O
seen	O
from	O
the	O
following	O
simple	O
derivation	O
:	O
</s>
<s>
Note	O
that	O
this	O
also	O
shows	O
that	O
the	O
Gramian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
definite	O
if	O
and	O
only	O
if	O
the	O
vectors	O
are	O
linearly	O
independent	O
(	O
that	O
is	O
,	O
for	O
all	O
)	O
.	O
</s>
<s>
Given	O
any	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
,	O
one	O
can	O
decompose	O
it	O
as	O
:	O
</s>
<s>
where	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
(	O
or	O
in	O
the	O
real	O
case	O
)	O
.	O
</s>
<s>
Thus	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	B-Algorithm
semidefinite	I-Algorithm
if	O
and	O
only	O
if	O
it	O
is	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
some	O
vectors	O
.	O
</s>
<s>
If	O
is	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
in	O
then	O
applying	O
any	O
rotation	O
or	O
reflection	O
of	O
(	O
any	O
orthogonal	O
transformation	O
,	O
that	O
is	O
,	O
any	O
Euclidean	B-Algorithm
isometry	I-Algorithm
preserving	O
0	O
)	O
to	O
the	O
sequence	O
of	O
vectors	O
results	O
in	O
the	O
same	O
Gram	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
dot	O
products	O
and	O
are	O
equal	O
if	O
and	O
only	O
if	O
some	O
rigid	B-Algorithm
transformation	I-Algorithm
of	O
transforms	O
the	O
vectors	O
to	O
and	O
0	O
to	O
0	O
.	O
</s>
<s>
The	O
same	O
holds	O
in	O
the	O
complex	O
case	O
,	O
with	O
unitary	B-Algorithm
transformations	I-Algorithm
in	O
place	O
of	O
orthogonal	O
ones	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
is	O
equal	O
to	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
in	O
then	O
there	O
is	O
a	O
unitary	B-Algorithm
matrix	O
(	O
meaning	O
)	O
such	O
that	O
for	O
.	O
</s>
<s>
That	O
is	O
,	O
a	O
real	O
or	O
complex	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
also	O
a	O
normal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
any	O
orthonormal	O
basis	O
is	O
the	O
identity	O
matrix	O
.	O
</s>
<s>
Equivalently	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
the	O
rows	O
or	O
the	O
columns	O
of	O
a	O
real	O
rotation	B-Algorithm
matrix	I-Algorithm
is	O
the	O
identity	O
matrix	O
.	O
</s>
<s>
Likewise	O
,	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
the	O
rows	O
or	O
columns	O
of	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
is	O
the	O
identity	O
matrix	O
.	O
</s>
<s>
The	O
rank	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
in	O
or	O
equals	O
the	O
dimension	O
of	O
the	O
space	O
spanned	O
by	O
these	O
vectors	O
.	O
</s>
<s>
The	O
Gram	O
determinant	O
or	O
Gramian	B-Algorithm
is	O
the	O
determinant	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
In	O
particular	O
,	O
the	O
vectors	O
are	O
linearly	O
independent	O
if	O
and	O
only	O
if	O
the	O
parallelotope	O
has	O
nonzero	O
n-dimensional	O
volume	O
,	O
if	O
and	O
only	O
if	O
Gram	O
determinant	O
is	O
nonzero	O
,	O
if	O
and	O
only	O
if	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
is	O
nonsingular	O
.	O
</s>
<s>
Indeed	O
,	O
is	O
Hermitian	B-Algorithm
,	O
and	O
so	O
can	O
be	O
decomposed	O
as	O
with	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
and	O
a	O
real	O
diagonal	O
matrix	O
.	O
</s>
