<s>
is	O
congruent	O
with	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
positive	O
real	O
entries	O
.	O
</s>
<s>
is	O
symmetric	B-Algorithm
or	O
Hermitian	O
,	O
and	O
all	O
its	O
eigenvalues	O
are	O
real	O
and	O
positive	O
.	O
</s>
<s>
is	O
symmetric	B-Algorithm
or	O
Hermitian	O
,	O
and	O
all	O
its	O
leading	O
principal	O
minors	O
are	O
positive	O
.	O
</s>
<s>
Some	O
authors	O
use	O
more	O
general	O
definitions	O
of	O
definiteness	O
,	O
including	O
some	O
non-symmetric	O
real	O
matrices	O
,	O
or	O
non-Hermitian	O
complex	O
ones	O
.	O
</s>
<s>
In	O
the	O
following	O
definitions	O
,	O
is	O
the	O
transpose	O
of	O
,	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
and	O
denotes	O
the	O
n-dimensional	O
zero-vector	O
.	O
</s>
<s>
An	O
symmetric	B-Algorithm
real	O
matrix	O
is	O
said	O
to	O
be	O
positive-definite	O
if	O
for	O
all	O
non-zero	O
in	O
.	O
</s>
<s>
An	O
symmetric	B-Algorithm
real	O
matrix	O
is	O
said	O
to	O
be	O
positive-semidefinite	O
or	O
non-negative-definite	O
if	O
for	O
all	O
in	O
.	O
</s>
<s>
An	O
symmetric	B-Algorithm
real	O
matrix	O
is	O
said	O
to	O
be	O
negative-definite	O
if	O
for	O
all	O
non-zero	O
in	O
.	O
</s>
<s>
An	O
symmetric	B-Algorithm
real	O
matrix	O
is	O
said	O
to	O
be	O
negative-semidefinite	O
or	O
non-positive-definite	O
if	O
for	O
all	O
in	O
.	O
</s>
<s>
An	O
symmetric	B-Algorithm
real	O
matrix	O
which	O
is	O
neither	O
positive	O
semidefinite	O
nor	O
negative	O
semidefinite	O
is	O
called	O
indefinite	O
.	O
</s>
<s>
its	O
transpose	O
is	O
equal	O
to	O
its	O
conjugate	O
)	O
,	O
since	O
being	O
real	O
,	O
it	O
equals	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
for	O
every	O
which	O
implies	O
.	O
</s>
<s>
By	O
this	O
definition	O
,	O
a	O
positive-definite	O
real	O
matrix	O
is	O
Hermitian	O
,	O
hence	O
symmetric	B-Algorithm
;	O
and	O
is	O
positive	O
for	O
all	O
non-zero	O
real	O
column	O
vectors	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
for	O
a	O
symmetric	B-Algorithm
real	O
matrix	O
,	O
the	O
condition	O
"	O
for	O
all	O
nonzero	O
real	O
vectors	O
"	O
does	O
imply	O
that	O
is	O
positive-definite	O
in	O
the	O
complex	O
sense	O
.	O
</s>
<s>
If	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
semi-definite	O
,	O
one	O
sometimes	O
writes	O
and	O
if	O
is	O
positive-definite	O
one	O
writes	O
.	O
</s>
<s>
The	O
notion	O
comes	O
from	O
functional	B-Application
analysis	I-Application
where	O
positive	O
semidefinite	O
matrices	O
define	O
positive	O
operators	O
.	O
</s>
<s>
This	O
may	O
be	O
confusing	O
,	O
as	O
sometimes	O
nonnegative	B-Algorithm
matrices	I-Algorithm
(	O
respectively	O
,	O
nonpositive	O
matrices	O
)	O
are	O
also	O
denoted	O
in	O
this	O
way	O
.	O
</s>
<s>
Let	O
be	O
an	O
Hermitian	B-Algorithm
matrix	I-Algorithm
(	O
this	O
includes	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
)	O
.	O
</s>
<s>
Let	O
be	O
an	O
eigendecomposition	O
of	O
,	O
where	O
is	O
a	O
unitary	B-Algorithm
complex	I-Algorithm
matrix	I-Algorithm
whose	O
columns	O
comprise	O
an	O
orthonormal	O
basis	O
of	O
eigenvectors	O
of	O
,	O
and	O
is	O
a	O
real	O
diagonal	B-Algorithm
matrix	I-Algorithm
whose	O
main	B-Algorithm
diagonal	I-Algorithm
contains	O
the	O
corresponding	O
eigenvalues	O
.	O
</s>
<s>
The	O
matrix	O
may	O
be	O
regarded	O
as	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
that	O
has	O
been	O
re-expressed	O
in	O
coordinates	O
of	O
the	O
(	O
eigenvectors	O
)	O
basis	O
.	O
</s>
<s>
Put	O
differently	O
,	O
applying	O
to	O
some	O
vector	O
,	O
giving	O
,	O
is	O
the	O
same	O
as	O
changing	O
the	O
basis	O
to	O
the	O
eigenvector	O
coordinate	O
system	O
using	O
,	O
giving	O
,	O
applying	O
the	O
stretching	B-Algorithm
transformation	I-Algorithm
to	O
the	O
result	O
,	O
giving	O
,	O
and	O
then	O
changing	O
the	O
basis	O
back	O
using	O
,	O
giving	O
.	O
</s>
<s>
For	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
,	O
this	O
is	O
true	O
only	O
if	O
each	O
element	O
of	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
—	O
that	O
is	O
,	O
every	O
eigenvalue	O
of	O
—	O
is	O
positive	O
.	O
</s>
<s>
Since	O
the	O
spectral	O
theorem	O
guarantees	O
all	O
eigenvalues	O
of	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
to	O
be	O
real	O
,	O
the	O
positivity	O
of	O
eigenvalues	O
can	O
be	O
checked	O
using	O
Descartes	O
 '	O
rule	O
of	O
alternating	O
signs	O
when	O
the	O
characteristic	O
polynomial	O
of	O
a	O
real	O
,	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
available	O
.	O
</s>
<s>
Let	O
be	O
an	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
of	O
a	O
matrix	O
with	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
In	O
other	O
words	O
,	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
semidefinite	O
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>
It	O
is	O
positive	O
definite	O
if	O
and	O
only	O
if	O
it	O
is	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
some	O
linearly	O
independent	O
vectors	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
rank	O
of	O
the	O
Gram	B-Algorithm
matrix	I-Algorithm
of	O
vectors	O
equals	O
the	O
dimension	O
of	O
the	O
space	O
spanned	O
by	O
these	O
vectors	O
.	O
</s>
<s>
if	O
for	O
some	O
matrix	O
and	O
if	O
is	O
any	O
unitary	B-Algorithm
matrix	O
(	O
meaning	O
)	O
,	O
</s>
<s>
However	O
,	O
this	O
is	O
the	O
only	O
way	O
in	O
which	O
two	O
decompositions	O
can	O
differ	O
:	O
the	O
decomposition	O
is	O
unique	O
up	O
to	O
unitary	B-Algorithm
transformations	I-Algorithm
.	O
</s>
<s>
When	O
this	O
means	O
is	O
unitary	B-Algorithm
.	O
</s>
<s>
A	O
real	O
unitary	B-Algorithm
matrix	I-Algorithm
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
,	O
which	O
describes	O
a	O
rigid	B-Algorithm
transformation	I-Algorithm
(	O
an	O
isometry	O
of	O
Euclidean	O
space	O
)	O
preserving	O
the	O
0	O
point	O
(	O
i.e.	O
</s>
<s>
rotations	B-Algorithm
and	O
reflections	B-Algorithm
,	O
without	O
translations	O
)	O
.	O
</s>
<s>
Therefore	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
(	O
and	O
0	O
to	O
0	O
)	O
.	O
</s>
<s>
A	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
semidefinite	O
if	O
and	O
only	O
if	O
there	O
is	O
a	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
(	O
in	O
particular	O
is	O
Hermitian	O
,	O
so	O
)	O
satisfying	O
.	O
</s>
<s>
A	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
can	O
be	O
written	O
as	O
,	O
where	O
is	O
lower	B-Algorithm
triangular	I-Algorithm
with	O
non-negative	O
diagonal	O
(	O
equivalently	O
where	O
is	O
upper	B-Algorithm
triangular	I-Algorithm
)	O
;	O
this	O
is	O
the	O
Cholesky	O
decomposition	O
.	O
</s>
<s>
Conversely	O
if	O
is	O
lower	B-Algorithm
triangular	I-Algorithm
with	O
nonnegative	O
diagonal	O
then	O
is	O
positive	O
semidefinite	O
.	O
</s>
<s>
Let	O
be	O
an	O
real	B-Algorithm
symmetric	I-Algorithm
matrix	I-Algorithm
,	O
and	O
let	O
be	O
the	O
"	O
unit	O
ball	O
"	O
defined	O
by	O
.	O
</s>
<s>
Let	O
be	O
an	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
associated	O
sesquilinear	B-Algorithm
form	I-Algorithm
is	O
an	O
inner	O
product	O
The	O
sesquilinear	B-Algorithm
form	I-Algorithm
defined	O
by	O
is	O
the	O
function	O
from	O
to	O
such	O
that	O
for	O
all	O
and	O
in	O
,	O
where	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
.	O
</s>
<s>
(	O
In	O
fact	O
,	O
every	O
inner	O
product	O
on	O
arises	O
in	O
this	O
fashion	O
from	O
a	O
Hermitian	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
.	O
)	O
</s>
<s>
This	O
condition	O
is	O
known	O
as	O
Sylvester	O
's	O
criterion	O
,	O
and	O
provides	O
an	O
efficient	O
test	O
of	O
positive	O
definiteness	O
of	O
a	O
symmetric	B-Algorithm
real	O
matrix	O
.	O
</s>
<s>
Namely	O
,	O
the	O
matrix	O
is	O
reduced	O
to	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
by	O
using	O
elementary	O
row	O
operations	O
,	O
as	O
in	O
the	O
first	O
part	O
of	O
the	O
Gaussian	B-Algorithm
elimination	I-Algorithm
method	O
,	O
taking	O
care	O
to	O
preserve	O
the	O
sign	O
of	O
its	O
determinant	O
during	O
pivoting	B-Algorithm
process	O
.	O
</s>
<s>
Since	O
the	O
kth	O
leading	O
principal	O
minor	O
of	O
a	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
the	O
product	O
of	O
its	O
diagonal	O
elements	O
up	O
to	O
row	O
,	O
Sylvester	O
's	O
criterion	O
is	O
equivalent	O
to	O
checking	O
whether	O
its	O
diagonal	O
elements	O
are	O
all	O
positive	O
.	O
</s>
<s>
This	O
condition	O
can	O
be	O
checked	O
each	O
time	O
a	O
new	O
row	O
of	O
the	O
triangular	B-Algorithm
matrix	I-Algorithm
is	O
obtained	O
.	O
</s>
<s>
A	O
positive	B-Algorithm
semidefinite	I-Algorithm
matrix	I-Algorithm
is	O
positive	O
definite	O
if	O
and	O
only	O
if	O
it	O
is	O
invertible	O
.	O
</s>
<s>
can	O
be	O
assumed	O
symmetric	B-Algorithm
by	O
replacing	O
it	O
with	O
.	O
</s>
<s>
A	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
definite	O
if	O
and	O
only	O
if	O
its	O
quadratic	O
form	O
is	O
a	O
strictly	O
convex	O
function	O
.	O
</s>
<s>
More	O
generally	O
,	O
any	O
quadratic	O
function	O
from	O
to	O
can	O
be	O
written	O
as	O
where	O
is	O
a	O
symmetric	B-Algorithm
matrix	O
,	O
is	O
a	O
real	O
-vector	O
,	O
and	O
a	O
real	O
constant	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
positive	B-Algorithm
definite	I-Algorithm
matrices	I-Algorithm
play	O
an	O
important	O
role	O
in	O
optimization	O
problems	O
.	O
</s>
<s>
One	O
symmetric	B-Algorithm
matrix	I-Algorithm
and	O
another	O
matrix	O
that	O
is	O
both	O
symmetric	B-Algorithm
and	O
positive	O
definite	O
can	O
be	O
simultaneously	O
diagonalized	O
.	O
</s>
<s>
This	O
is	O
so	O
although	O
simultaneous	O
diagonalization	O
is	O
not	O
necessarily	O
performed	O
with	O
a	O
similarity	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Let	O
be	O
a	O
symmetric	B-Algorithm
and	O
a	O
symmetric	B-Algorithm
and	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Manipulation	O
now	O
yields	O
where	O
is	O
a	O
matrix	O
having	O
as	O
columns	O
the	O
generalized	O
eigenvectors	O
and	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
of	O
the	O
generalized	O
eigenvalues	O
.	O
</s>
<s>
Note	O
that	O
this	O
result	O
does	O
not	O
contradict	O
what	O
is	O
said	O
on	O
simultaneous	O
diagonalization	O
in	O
the	O
article	O
Diagonalizable	O
matrix	O
,	O
which	O
refers	O
to	O
simultaneous	O
diagonalization	O
by	O
a	O
similarity	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Every	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
is	O
invertible	O
and	O
its	O
inverse	O
is	O
also	O
positive	O
definite	O
.	O
</s>
<s>
The	O
diagonal	O
entries	O
of	O
a	O
positive-semidefinite	B-Algorithm
matrix	I-Algorithm
are	O
real	O
and	O
non-negative	O
.	O
</s>
<s>
An	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
definite	O
if	O
it	O
satisfies	O
the	O
following	O
trace	O
inequalities	O
:	O
</s>
<s>
The	O
set	O
of	O
positive	O
semidefinite	O
symmetric	B-Algorithm
matrices	I-Algorithm
is	O
convex	O
.	O
</s>
<s>
If	O
is	O
a	O
symmetric	B-Algorithm
Toeplitz	B-Algorithm
matrix	I-Algorithm
,	O
i.e.	O
</s>
<s>
If	O
is	O
real	O
,	O
then	O
there	O
is	O
a	O
such	O
that	O
,	O
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
is	O
a	O
real	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
,	O
then	O
there	O
exists	O
a	O
positive	O
real	O
number	O
such	O
that	O
for	O
every	O
vector	O
,	O
.	O
</s>
<s>
A	O
Hermitian	B-Algorithm
matrix	I-Algorithm
is	O
positive	O
semidefinite	O
if	O
and	O
only	O
if	O
all	O
of	O
its	O
principal	O
minors	O
are	O
nonnegative	O
.	O
</s>
<s>
It	O
is	O
however	O
not	O
enough	O
to	O
consider	O
the	O
leading	O
principal	O
minors	O
only	O
,	O
as	O
is	O
checked	O
on	O
the	O
diagonal	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
0	O
and	O
−1	O
.	O
</s>
<s>
A	O
positive	O
matrix	O
may	O
also	O
be	O
defined	O
by	O
blocks	B-Algorithm
:	O
</s>
<s>
Converse	O
results	O
can	O
be	O
proved	O
with	O
stronger	O
conditions	O
on	O
the	O
blocks	B-Algorithm
,	O
for	O
instance	O
,	O
using	O
the	O
Schur	O
complement	O
.	O
</s>
<s>
A	O
general	O
quadratic	O
form	O
on	O
real	O
variables	O
can	O
always	O
be	O
written	O
as	O
where	O
is	O
the	O
column	O
vector	O
with	O
those	O
variables	O
,	O
and	O
is	O
a	O
symmetric	B-Algorithm
real	O
matrix	O
.	O
</s>
<s>
Conversely	O
,	O
every	O
positive	B-Algorithm
semi-definite	I-Algorithm
matrix	I-Algorithm
is	O
the	O
covariance	O
matrix	O
of	O
some	O
multivariate	O
distribution	O
.	O
</s>
<s>
real	O
non-symmetric	O
)	O
as	O
positive	O
definite	O
if	O
for	O
all	O
non-zero	O
complex	O
vectors	O
,	O
where	O
denotes	O
the	O
real	O
part	O
of	O
a	O
complex	O
number	O
.	O
</s>
<s>
Similarly	O
,	O
if	O
and	O
are	O
real	O
,	O
we	O
have	O
for	O
all	O
real	O
nonzero	O
vectors	O
if	O
and	O
only	O
if	O
the	O
symmetric	B-Algorithm
part	O
is	O
positive	O
definite	O
in	O
the	O
narrower	O
sense	O
.	O
</s>
<s>
Consequently	O
,	O
a	O
non-symmetric	O
real	O
matrix	O
with	O
only	O
positive	O
eigenvalues	O
does	O
not	O
need	O
to	O
be	O
positive	O
definite	O
.	O
</s>
<s>
Fourier	O
's	O
law	O
of	O
heat	O
conduction	O
,	O
giving	O
heat	O
flux	O
in	O
terms	O
of	O
the	O
temperature	O
gradient	O
is	O
written	O
for	O
anisotropic	O
media	O
as	O
,	O
in	O
which	O
is	O
the	O
symmetric	B-Algorithm
thermal	O
conductivity	O
matrix	O
.	O
</s>
