<s>
Without	O
further	O
specifications	O
,	O
matrices	O
represent	O
linear	B-Architecture
maps	I-Architecture
,	O
and	O
allow	O
explicit	O
computations	O
in	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
Therefore	O
,	O
the	O
study	O
of	O
matrices	O
is	O
a	O
large	O
part	O
of	O
linear	B-Language
algebra	I-Language
,	O
and	O
most	O
properties	O
and	O
operations	O
of	O
abstract	O
linear	B-Language
algebra	I-Language
can	O
be	O
expressed	O
in	O
terms	O
of	O
matrices	O
.	O
</s>
<s>
For	O
example	O
,	O
matrix	O
multiplication	O
represents	O
composition	B-Application
of	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
Not	O
all	O
matrices	O
are	O
related	O
to	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
This	O
is	O
,	O
in	O
particular	O
,	O
the	O
case	O
in	O
graph	O
theory	O
,	O
of	O
incidence	B-Algorithm
matrices	I-Algorithm
,	O
and	O
adjacency	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
This	O
article	O
focuses	O
on	O
matrices	O
related	O
to	O
linear	B-Language
algebra	I-Language
,	O
and	O
,	O
unless	O
otherwise	O
specified	O
,	O
all	O
matrices	O
represent	O
linear	B-Architecture
maps	I-Architecture
or	O
may	O
be	O
viewed	O
as	O
such	O
.	O
</s>
<s>
Square	B-Algorithm
matrices	I-Algorithm
,	O
matrices	O
with	O
the	O
same	O
number	O
of	O
rows	O
and	O
columns	O
,	O
play	O
a	O
major	O
role	O
in	O
matrix	B-Architecture
theory	I-Architecture
.	O
</s>
<s>
Square	B-Algorithm
matrices	I-Algorithm
of	O
a	O
given	O
dimension	O
form	O
a	O
noncommutative	O
ring	O
,	O
which	O
is	O
one	O
of	O
the	O
most	O
common	O
examples	O
of	O
a	O
noncommutative	O
ring	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
a	O
number	O
associated	O
to	O
the	O
matrix	O
,	O
which	O
is	O
fundamental	O
for	O
the	O
study	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
;	O
for	O
example	O
,	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
invertible	O
if	O
and	O
only	O
if	O
it	O
has	O
a	O
nonzero	O
determinant	O
,	O
and	O
the	O
eigenvalues	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
are	O
the	O
roots	O
of	O
a	O
polynomial	O
determinant	O
.	O
</s>
<s>
In	O
geometry	O
,	O
matrices	O
are	O
widely	O
used	O
for	O
specifying	O
and	O
representing	O
geometric	B-Algorithm
transformations	I-Algorithm
(	O
for	O
example	O
rotations	O
)	O
and	O
coordinate	O
changes	O
.	O
</s>
<s>
In	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
many	O
computational	O
problems	O
are	O
solved	O
by	O
reducing	O
them	O
to	O
a	O
matrix	O
computation	O
,	O
and	O
this	O
often	O
involves	O
computing	O
with	O
matrices	O
of	O
huge	O
dimension	O
.	O
</s>
<s>
Matrices	O
are	O
used	O
in	O
most	O
areas	O
of	O
mathematics	O
and	O
most	O
scientific	O
fields	O
,	O
either	O
directly	O
,	O
or	O
through	O
their	O
use	O
in	O
geometry	O
and	O
numerical	B-General_Concept
analysis	I-General_Concept
.	O
</s>
<s>
Matrix	B-Architecture
theory	I-Architecture
is	O
the	O
branch	O
of	O
mathematics	O
that	O
focuses	O
on	O
the	O
study	O
of	O
matrices	O
.	O
</s>
<s>
It	O
was	O
initially	O
a	O
sub-branch	O
of	O
linear	B-Language
algebra	I-Language
,	O
but	O
soon	O
grew	O
to	O
include	O
subjects	O
related	O
to	O
graph	O
theory	O
,	O
algebra	O
,	O
combinatorics	O
and	O
statistics	O
.	O
</s>
<s>
A	O
matrix	O
with	O
the	O
same	O
number	O
of	O
rows	O
and	O
columns	O
is	O
called	O
a	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
some	O
contexts	O
,	O
such	O
as	O
computer	B-General_Concept
algebra	I-General_Concept
programs	I-General_Concept
,	O
it	O
is	O
useful	O
to	O
consider	O
a	O
matrix	O
with	O
no	O
rows	O
or	O
no	O
columns	O
,	O
called	O
an	O
empty	O
matrix	O
.	O
</s>
<s>
+Overview	O
of	O
a	O
matrix	O
size	O
Name	O
Size	O
Example	O
Description	O
Row	O
vector	O
1×n	O
A	O
matrix	O
with	O
one	O
row	O
,	O
sometimes	O
used	O
to	O
represent	O
a	O
vector	O
Column	O
vector	O
n×1	O
A	O
matrix	O
with	O
one	O
column	O
,	O
sometimes	O
used	O
to	O
represent	O
a	O
vector	O
Square	B-Algorithm
matrix	I-Algorithm
n×n	O
A	O
matrix	O
with	O
the	O
same	O
number	O
of	O
rows	O
and	O
columns	O
,	O
sometimes	O
used	O
to	O
represent	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
a	O
vector	O
space	O
to	O
itself	O
,	O
such	O
as	O
reflection	B-Algorithm
,	O
rotation	B-General_Concept
,	O
or	O
shearing	B-Algorithm
.	O
</s>
<s>
An	O
asterisk	B-Language
is	O
occasionally	O
used	O
to	O
refer	O
to	O
whole	O
rows	O
or	O
columns	O
in	O
a	O
matrix	O
.	O
</s>
<s>
They	O
arise	O
in	O
solving	O
matrix	B-Architecture
equations	I-Architecture
such	O
as	O
the	O
Sylvester	B-Algorithm
equation	I-Algorithm
.	O
</s>
<s>
If	O
A	O
has	O
no	O
inverse	O
,	O
solutions	O
—	O
if	O
any	O
—	O
can	O
be	O
found	O
using	O
its	O
generalized	B-Algorithm
inverse	I-Algorithm
.	O
</s>
<s>
Matrices	O
and	O
matrix	O
multiplication	O
reveal	O
their	O
essential	O
features	O
when	O
related	O
to	O
linear	B-Architecture
transformations	I-Architecture
,	O
also	O
known	O
as	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
A	O
real	O
m-by-n	O
matrix	O
A	O
gives	O
rise	O
to	O
a	O
linear	B-Architecture
transformation	I-Architecture
R	O
→	O
R	O
mapping	O
each	O
vector	O
x	O
in	O
R	O
to	O
the	O
(	O
matrix	O
)	O
product	O
Ax	O
,	O
which	O
is	O
a	O
vector	O
in	O
R	O
.	O
Conversely	O
,	O
each	O
linear	B-Architecture
transformation	I-Architecture
f	O
:	O
R	O
→	O
R	O
arises	O
from	O
a	O
unique	O
m-by-n	O
matrix	O
A	O
:	O
explicitly	O
,	O
the	O
of	O
A	O
is	O
the	O
i	O
coordinate	O
of	O
f(e )	O
,	O
where	O
e	O
=	O
(	O
0	O
,...,	O
0	O
,	O
1	O
,	O
0	O
,...,	O
0	O
)	O
is	O
the	O
unit	O
vector	O
with	O
1	O
in	O
the	O
j	O
position	O
and	O
0	O
elsewhere	O
.	O
</s>
<s>
The	O
matrix	O
A	O
is	O
said	O
to	O
represent	O
the	O
linear	B-Architecture
map	I-Architecture
f	O
,	O
and	O
A	O
is	O
called	O
the	O
transformation	O
matrix	O
of	O
f	O
.	O
</s>
<s>
The	O
following	O
table	O
shows	O
several	O
2×2	O
real	B-Architecture
matrices	I-Architecture
with	O
the	O
associated	O
linear	B-Architecture
maps	I-Architecture
of	O
R	O
.	O
The	O
original	O
is	O
mapped	O
to	O
the	O
grid	O
and	O
shapes	O
.	O
</s>
<s>
Equivalently	O
it	O
is	O
the	O
dimension	O
of	O
the	O
image	O
of	O
the	O
linear	B-Architecture
map	I-Architecture
represented	O
by	O
A	O
.	O
</s>
<s>
The	O
rank	O
–	O
nullity	O
theorem	O
states	O
that	O
the	O
dimension	O
of	O
the	O
kernel	B-Algorithm
of	I-Algorithm
a	I-Algorithm
matrix	I-Algorithm
plus	O
the	O
rank	O
equals	O
the	O
number	O
of	O
columns	O
of	O
the	O
matrix	O
.	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	O
with	O
the	O
same	O
number	O
of	O
rows	O
and	O
columns	O
.	O
</s>
<s>
An	O
n-by-n	O
matrix	O
is	O
known	O
as	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n	O
.	O
Any	O
two	O
square	B-Algorithm
matrices	I-Algorithm
of	O
the	O
same	O
order	O
can	O
be	O
added	O
and	O
multiplied	O
.	O
</s>
<s>
The	O
entries	O
a	O
form	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
all	O
entries	O
of	O
A	O
below	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
,	O
A	O
is	O
called	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Similarly	O
if	O
all	O
entries	O
of	O
A	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
,	O
A	O
is	O
called	O
a	O
lower	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
all	O
entries	O
outside	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
,	O
A	O
is	O
called	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrix	I-Algorithm
I	O
of	O
size	O
n	O
is	O
the	O
n-by-n	O
matrix	O
in	O
which	O
all	O
the	O
elements	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
equal	O
to	O
1	O
and	O
all	O
other	O
elements	O
are	O
equal	O
to	O
0	O
,	O
for	O
example	O
,	O
</s>
<s>
It	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
order	O
n	O
,	O
and	O
also	O
a	O
special	O
kind	O
of	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
is	O
called	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
because	O
multiplication	O
with	O
it	O
leaves	O
a	O
matrix	O
unchanged	O
:	O
</s>
<s>
A	O
nonzero	O
scalar	O
multiple	O
of	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
is	O
called	O
a	O
scalar	O
matrix	O
.	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
A	O
that	O
is	O
equal	O
to	O
its	O
transpose	O
,	O
that	O
is	O
,	O
A	O
=	O
A	O
,	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
instead	O
,	O
A	O
is	O
equal	O
to	O
the	O
negative	O
of	O
its	O
transpose	O
,	O
that	O
is	O
,	O
A	O
=	O
−A	O
,	O
then	O
A	O
is	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
complex	O
matrices	O
,	O
symmetry	O
is	O
often	O
replaced	O
by	O
the	O
concept	O
of	O
Hermitian	B-Algorithm
matrices	I-Algorithm
,	O
which	O
satisfy	O
A	O
=	O
A	O
,	O
where	O
the	O
star	O
or	O
asterisk	B-Language
denotes	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
the	O
matrix	O
,	O
that	O
is	O
,	O
the	O
transpose	O
of	O
the	O
complex	O
conjugate	O
of	O
A	O
.	O
</s>
<s>
By	O
the	O
spectral	O
theorem	O
,	O
real	O
symmetric	B-Algorithm
matrices	I-Algorithm
and	O
complex	O
Hermitian	B-Algorithm
matrices	I-Algorithm
have	O
an	O
eigenbasis	O
;	O
that	O
is	O
,	O
every	O
vector	O
is	O
expressible	O
as	O
a	O
linear	O
combination	O
of	O
eigenvectors	O
.	O
</s>
<s>
where	O
I	O
is	O
the	O
n×n	O
identity	B-Algorithm
matrix	I-Algorithm
with	O
1s	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
0s	O
elsewhere	O
.	O
</s>
<s>
Positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
Indefinite	B-Algorithm
matrix	I-Algorithm
Q(x, y )	O
=	O
x	O
+	O
y	O
Q(x, y )	O
=	O
x	O
−	O
y	O
150px	O
Points	O
such	O
that	O
Q(x, y )	O
=	O
1	O
(	O
Ellipse	O
)	O
.	O
</s>
<s>
If	O
only	O
yields	O
negative	O
values	O
then	O
is	O
negative-definite	O
;	O
if	O
does	O
produce	O
both	O
negative	O
and	O
positive	O
values	O
then	O
is	O
indefinite	B-Algorithm
.	O
</s>
<s>
If	O
the	O
quadratic	O
form	O
yields	O
only	O
non-negative	O
values	O
(	O
positive	O
or	O
zero	O
)	O
,	O
the	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
called	O
positive-semidefinite	O
(	O
or	O
if	O
only	O
non-positive	O
values	O
,	O
then	O
negative-semidefinite	O
)	O
;	O
hence	O
the	O
matrix	O
is	O
indefinite	B-Algorithm
precisely	O
when	O
it	O
is	O
neither	O
positive-semidefinite	O
nor	O
negative-semidefinite	O
.	O
</s>
<s>
A	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
positive-definite	O
if	O
and	O
only	O
if	O
all	O
its	O
eigenvalues	O
are	O
positive	O
,	O
that	O
is	O
,	O
the	O
matrix	O
is	O
positive-semidefinite	O
and	O
it	O
is	O
invertible	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
complex	O
matrices	O
,	O
the	O
same	O
terminology	O
and	O
result	O
apply	O
,	O
with	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
quadratic	O
form	O
,	O
bilinear	O
form	O
,	O
and	O
transpose	O
replaced	O
respectively	O
by	O
Hermitian	B-Algorithm
matrix	I-Algorithm
,	O
Hermitian	O
form	O
,	O
sesquilinear	B-Algorithm
form	I-Algorithm
,	O
and	O
conjugate	B-Algorithm
transpose	I-Algorithm
.	O
</s>
<s>
An	O
orthogonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
with	O
real	O
entries	O
whose	O
columns	O
and	O
rows	O
are	O
orthogonal	O
unit	O
vectors	O
(	O
that	O
is	O
,	O
orthonormal	B-Algorithm
vectors	I-Algorithm
)	O
.	O
</s>
<s>
where	O
I	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
size	O
n	O
.	O
</s>
<s>
An	O
orthogonal	B-Algorithm
matrix	I-Algorithm
A	O
is	O
necessarily	O
invertible	O
(	O
with	O
inverse	O
)	O
,	O
unitary	B-Algorithm
(	O
)	O
,	O
and	O
normal	B-Algorithm
(	O
)	O
.	O
</s>
<s>
The	O
determinant	O
of	O
any	O
orthogonal	B-Algorithm
matrix	I-Algorithm
is	O
either	O
or	O
.	O
</s>
<s>
A	O
special	B-Algorithm
orthogonal	I-Algorithm
matrix	I-Algorithm
is	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
with	O
determinant	O
+1	O
.	O
</s>
<s>
As	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
every	O
orthogonal	B-Algorithm
matrix	I-Algorithm
with	O
determinant	O
is	O
a	O
pure	O
rotation	B-General_Concept
without	O
reflection	B-Algorithm
,	O
i.e.	O
,	O
the	O
transformation	O
preserves	O
the	O
orientation	O
of	O
the	O
transformed	O
structure	O
,	O
while	O
every	O
orthogonal	B-Algorithm
matrix	I-Algorithm
with	O
determinant	O
reverses	O
the	O
orientation	O
,	O
i.e.	O
,	O
is	O
a	O
composition	B-Application
of	O
a	O
pure	O
reflection	B-Algorithm
and	O
a	O
(	O
possibly	O
null	O
)	O
rotation	B-General_Concept
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrices	I-Algorithm
have	O
determinant	O
,	O
and	O
are	O
pure	O
rotations	O
by	O
an	O
angle	O
zero	O
.	O
</s>
<s>
The	O
complex	O
analogue	O
of	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
is	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
trace	O
,	O
tr(A )	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
A	O
is	O
the	O
sum	O
of	O
its	O
diagonal	O
entries	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
trace	O
of	O
the	O
product	O
of	O
more	O
than	O
two	O
matrices	O
is	O
independent	O
of	O
cyclic	B-Algorithm
permutations	I-Algorithm
of	O
the	O
matrices	O
,	O
however	O
this	O
does	O
not	O
in	O
general	O
apply	O
for	O
arbitrary	O
permutations	O
(	O
for	O
example	O
,	O
tr(ABC )	O
≠	O
tr(BAC )	O
,	O
in	O
general	O
)	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
A	O
(	O
denoted	O
det(A )	O
or	O
|A|	O
)	O
is	O
a	O
number	O
encoding	O
certain	O
properties	O
of	O
the	O
matrix	O
.	O
</s>
<s>
Its	O
absolute	O
value	O
equals	O
the	O
area	O
(	O
in	O
R	O
)	O
or	O
volume	O
(	O
in	O
R	O
)	O
of	O
the	O
image	O
of	O
the	O
unit	O
square	O
(	O
or	O
cube	O
)	O
,	O
while	O
its	O
sign	O
corresponds	O
to	O
the	O
orientation	O
of	O
the	O
corresponding	O
linear	B-Architecture
map	I-Architecture
:	O
the	O
determinant	O
is	O
positive	O
if	O
and	O
only	O
if	O
the	O
orientation	O
is	O
preserved	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
product	O
of	O
square	B-Algorithm
matrices	I-Algorithm
equals	O
the	O
product	O
of	O
their	O
determinants	O
:	O
</s>
<s>
Using	O
these	O
operations	O
,	O
any	O
matrix	O
can	O
be	O
transformed	O
to	O
a	O
lower	O
(	O
or	O
upper	O
)	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
and	O
for	O
such	O
matrices	O
,	O
the	O
determinant	O
equals	O
the	O
product	O
of	O
the	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
;	O
this	O
provides	O
a	O
method	O
to	O
calculate	O
the	O
determinant	O
of	O
any	O
matrix	O
.	O
</s>
<s>
Determinants	O
can	O
be	O
used	O
to	O
solve	O
linear	O
systems	O
using	O
Cramer	O
's	O
rule	O
,	O
where	O
the	O
division	O
of	O
the	O
determinants	O
of	O
two	O
related	O
square	B-Algorithm
matrices	I-Algorithm
equates	O
to	O
the	O
value	O
of	O
each	O
of	O
the	O
system	O
's	O
variables	O
.	O
</s>
<s>
According	O
to	O
the	O
Cayley	O
–	O
Hamilton	O
theorem	O
,	O
p(A )	O
=	O
0	O
,	O
that	O
is	O
,	O
the	O
result	O
of	O
substituting	O
the	O
matrix	O
itself	O
into	O
its	O
own	O
characteristic	O
polynomial	O
yields	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
eigenvectors	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
can	O
be	O
obtained	O
by	O
finding	O
a	O
sequence	O
of	O
vectors	O
x	O
converging	B-Algorithm
to	O
an	O
eigenvector	O
when	O
n	O
tends	O
to	O
infinity	B-Application
.	O
</s>
<s>
The	O
domain	O
studying	O
these	O
matters	O
is	O
called	O
numerical	O
linear	B-Language
algebra	I-Language
.	O
</s>
<s>
As	O
with	O
other	O
numerical	O
situations	O
,	O
two	O
main	O
aspects	O
are	O
the	O
complexity	B-General_Concept
of	O
algorithms	O
and	O
their	O
numerical	B-Algorithm
stability	I-Algorithm
.	O
</s>
<s>
Determining	O
the	O
complexity	B-General_Concept
of	O
an	O
algorithm	O
means	O
finding	O
upper	O
bounds	O
or	O
estimates	O
of	O
how	O
many	O
elementary	O
operations	O
such	O
as	O
additions	O
and	O
multiplications	O
of	O
scalars	O
are	O
necessary	O
to	O
perform	O
some	O
algorithm	O
,	O
for	O
example	O
,	O
multiplication	O
of	O
matrices	O
.	O
</s>
<s>
The	O
Strassen	B-Algorithm
algorithm	I-Algorithm
outperforms	O
this	O
"	O
naive	O
"	O
algorithm	O
;	O
it	O
needs	O
only	O
n	O
multiplications	O
.	O
</s>
<s>
An	O
important	O
case	O
are	O
sparse	B-Algorithm
matrices	I-Algorithm
,	O
that	O
is	O
,	O
matrices	O
most	O
of	O
whose	O
entries	O
are	O
zero	O
.	O
</s>
<s>
There	O
are	O
specifically	O
adapted	O
algorithms	O
for	O
,	O
say	O
,	O
solving	O
linear	O
systems	O
Ax	O
=	O
b	O
for	O
sparse	B-Algorithm
matrices	I-Algorithm
A	O
,	O
such	O
as	O
the	O
conjugate	B-Algorithm
gradient	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
An	O
algorithm	O
is	O
,	O
roughly	O
speaking	O
,	O
numerically	B-Algorithm
stable	I-Algorithm
,	O
if	O
little	O
deviations	O
in	O
the	O
input	O
values	O
do	O
not	O
lead	O
to	O
big	O
deviations	O
in	O
the	O
result	O
.	O
</s>
<s>
The	O
norm	O
of	O
a	O
matrix	O
can	O
be	O
used	O
to	O
capture	O
the	O
conditioning	B-Algorithm
of	O
linear	O
algebraic	O
problems	O
,	O
such	O
as	O
computing	O
a	O
matrix	O
's	O
inverse	O
.	O
</s>
<s>
The	O
original	O
Dartmouth	B-Language
BASIC	I-Language
had	O
built-in	O
commands	O
for	O
matrix	O
arithmetic	O
on	O
arrays	O
from	O
its	O
second	O
edition	O
implementation	O
in	O
1964	O
.	O
</s>
<s>
As	O
early	O
as	O
the	O
1970s	O
,	O
some	O
engineering	O
desktop	O
computers	O
such	O
as	O
the	O
HP	B-Device
9830	I-Device
had	O
ROM	O
cartridges	O
to	O
add	O
BASIC	O
commands	O
for	O
matrices	O
.	O
</s>
<s>
Some	O
computer	O
languages	O
such	O
as	O
APL	B-Language
were	O
designed	O
to	O
manipulate	O
matrices	O
,	O
and	O
various	B-Application
mathematical	I-Application
programs	I-Application
can	O
be	O
used	O
to	O
aid	O
computing	O
with	O
matrices	O
.	O
</s>
<s>
The	O
LU	O
decomposition	O
factors	O
matrices	O
as	O
a	O
product	O
of	O
lower	O
(	O
L	O
)	O
and	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrices	I-Algorithm
(	O
U	O
)	O
.	O
</s>
<s>
Likewise	O
,	O
inverses	O
of	O
triangular	B-Algorithm
matrices	I-Algorithm
are	O
algorithmically	O
easier	O
to	O
calculate	O
.	O
</s>
<s>
The	O
Gaussian	B-Algorithm
elimination	I-Algorithm
is	O
a	O
similar	O
algorithm	O
;	O
it	O
transforms	O
any	O
matrix	O
to	O
row	O
echelon	O
form	O
.	O
</s>
<s>
Both	O
methods	O
proceed	O
by	O
multiplying	O
the	O
matrix	O
by	O
suitable	O
elementary	O
matrices	O
,	O
which	O
correspond	O
to	O
permuting	B-Algorithm
rows	I-Algorithm
or	I-Algorithm
columns	I-Algorithm
and	O
adding	O
multiples	O
of	O
one	O
row	O
to	O
another	O
row	O
.	O
</s>
<s>
Singular	O
value	O
decomposition	O
expresses	O
any	O
matrix	O
A	O
as	O
a	O
product	O
UDV	O
,	O
where	O
U	O
and	O
V	O
are	O
unitary	B-Algorithm
matrices	I-Algorithm
and	O
D	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
eigendecomposition	O
or	O
diagonalization	O
expresses	O
A	O
as	O
a	O
product	O
VDV	O
,	O
where	O
D	O
is	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
and	O
V	O
is	O
a	O
suitable	O
invertible	O
matrix	O
.	O
</s>
<s>
If	O
A	O
can	O
be	O
written	O
in	O
this	O
form	O
,	O
it	O
is	O
called	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
More	O
generally	O
,	O
and	O
applicable	O
to	O
all	O
matrices	O
,	O
the	O
Jordan	O
decomposition	O
transforms	O
a	O
matrix	O
into	O
Jordan	O
normal	B-Algorithm
form	O
,	O
that	O
is	O
to	O
say	O
matrices	O
whose	O
only	O
nonzero	O
entries	O
are	O
the	O
eigenvalues	O
λ	O
to	O
λ	O
of	O
A	O
,	O
placed	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
possibly	O
entries	O
equal	O
to	O
one	O
directly	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
,	O
as	O
shown	O
at	O
the	O
right	O
.	O
</s>
<s>
and	O
the	O
power	O
of	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
can	O
be	O
calculated	O
by	O
taking	O
the	O
corresponding	O
powers	O
of	O
the	O
diagonal	O
entries	O
,	O
which	O
is	O
much	O
easier	O
than	O
doing	O
the	O
exponentiation	O
for	O
A	O
instead	O
.	O
</s>
<s>
To	O
avoid	O
numerically	B-General_Concept
ill-conditioned	B-Algorithm
situations	O
,	O
further	O
algorithms	O
such	O
as	O
the	O
Schur	O
decomposition	O
can	O
be	O
employed	O
.	O
</s>
<s>
Abstract	O
algebra	O
uses	O
matrices	O
with	O
entries	O
in	O
more	O
general	O
fields	O
or	O
even	O
rings	O
,	O
while	O
linear	B-Language
algebra	I-Language
codifies	O
properties	O
of	O
matrices	O
in	O
the	O
notion	O
of	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
Another	O
extension	O
is	O
tensors	B-Device
,	O
which	O
can	O
be	O
seen	O
as	O
higher-dimensional	O
arrays	O
of	O
numbers	O
,	O
as	O
opposed	O
to	O
vectors	O
,	O
which	O
can	O
often	O
be	O
realized	O
as	O
sequences	O
of	O
numbers	O
,	O
while	O
matrices	O
are	O
rectangular	O
or	O
two-dimensional	O
arrays	O
of	O
numbers	O
.	O
</s>
<s>
For	O
example	O
,	O
coding	B-Error_Name
theory	I-Error_Name
makes	O
use	O
of	O
matrices	O
over	O
finite	O
fields	O
.	O
</s>
<s>
The	O
possibility	O
to	O
reinterpret	O
the	O
entries	O
of	O
a	O
matrix	O
as	O
elements	O
of	O
a	O
larger	O
field	O
(	O
for	O
example	O
,	O
to	O
view	O
a	O
real	O
matrix	O
as	O
a	O
complex	O
matrix	O
whose	O
entries	O
happen	O
to	O
be	O
all	O
real	O
)	O
then	O
allows	O
considering	O
each	O
square	B-Algorithm
matrix	I-Algorithm
to	O
possess	O
a	O
full	O
set	O
of	O
eigenvalues	O
.	O
</s>
<s>
The	O
set	O
M(n, R )	O
(	O
also	O
denoted	O
Mn(R )	O
)	O
of	O
all	O
square	O
n-by-n	O
matrices	O
over	O
R	O
is	O
a	O
ring	O
called	O
matrix	O
ring	O
,	O
isomorphic	O
to	O
the	O
endomorphism	O
ring	O
of	O
the	O
left	O
R-module	O
R	O
.	O
If	O
the	O
ring	O
R	O
is	O
commutative	O
,	O
that	O
is	O
,	O
its	O
multiplication	O
is	O
commutative	O
,	O
then	O
the	O
ring	O
M(n, R )	O
is	O
also	O
an	O
associative	O
algebra	O
over	O
R	O
.	O
The	O
determinant	O
of	O
square	B-Algorithm
matrices	I-Algorithm
over	O
a	O
commutative	O
ring	O
R	O
can	O
still	O
be	O
defined	O
using	O
the	O
Leibniz	O
formula	O
;	O
such	O
a	O
matrix	O
is	O
invertible	O
if	O
and	O
only	O
if	O
its	O
determinant	O
is	O
invertible	O
in	O
R	O
,	O
generalising	O
the	O
situation	O
over	O
a	O
field	O
F	O
,	O
where	O
every	O
nonzero	O
element	O
is	O
invertible	O
.	O
</s>
<s>
Matrices	O
over	O
superrings	O
are	O
called	O
supermatrices	B-Algorithm
.	O
</s>
<s>
One	O
special	O
but	O
common	O
case	O
is	O
block	B-Algorithm
matrices	I-Algorithm
,	O
which	O
may	O
be	O
considered	O
as	O
matrices	O
whose	O
entries	O
themselves	O
are	O
matrices	O
.	O
</s>
<s>
The	O
entries	O
need	O
not	O
be	O
square	B-Algorithm
matrices	I-Algorithm
,	O
and	O
thus	O
need	O
not	O
be	O
members	O
of	O
any	O
ring	O
;	O
but	O
their	O
sizes	O
must	O
fulfill	O
certain	O
compatibility	O
conditions	O
.	O
</s>
<s>
Linear	B-Architecture
maps	I-Architecture
R	O
→	O
R	O
are	O
equivalent	O
to	O
m-by-n	O
matrices	O
,	O
as	O
described	O
above	O
.	O
</s>
<s>
The	O
matrix	O
depends	O
on	O
the	O
choice	O
of	O
the	O
bases	O
:	O
different	O
choices	O
of	O
bases	O
give	O
rise	O
to	O
different	O
,	O
but	O
equivalent	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Many	O
of	O
the	O
above	O
concrete	O
notions	O
can	O
be	O
reinterpreted	O
in	O
this	O
light	O
,	O
for	O
example	O
,	O
the	O
transpose	O
matrix	O
A	O
describes	O
the	O
transpose	B-Algorithm
of	I-Algorithm
the	I-Algorithm
linear	I-Algorithm
map	I-Algorithm
given	O
by	O
A	O
,	O
with	O
respect	O
to	O
the	O
dual	O
bases	O
.	O
</s>
<s>
These	O
properties	O
can	O
be	O
restated	O
more	O
naturally	O
:	O
the	O
category	O
of	O
all	O
matrices	O
with	O
entries	O
in	O
a	O
field	O
with	O
multiplication	O
as	O
composition	B-Application
is	O
equivalent	O
to	O
the	O
category	O
of	O
finite-dimensional	O
vector	O
spaces	O
and	O
linear	B-Architecture
maps	I-Architecture
over	O
this	O
field	O
.	O
</s>
<s>
When	O
n	O
=	O
m	O
composition	B-Application
of	O
these	O
maps	O
is	O
possible	O
,	O
and	O
this	O
gives	O
rise	O
to	O
the	O
matrix	O
ring	O
of	O
n×n	O
matrices	O
representing	O
the	O
endomorphism	O
ring	O
of	O
R	O
.	O
</s>
<s>
Every	O
orthogonal	B-Algorithm
matrix	I-Algorithm
has	O
determinant	O
1	O
or	O
−1	O
.	O
</s>
<s>
Orthogonal	B-Algorithm
matrices	I-Algorithm
with	O
determinant	O
1	O
form	O
a	O
subgroup	O
called	O
special	O
orthogonal	O
group	O
.	O
</s>
<s>
Every	O
finite	O
group	O
is	O
isomorphic	O
to	O
a	O
matrix	O
group	O
,	O
as	O
one	O
can	O
see	O
by	O
considering	O
the	O
regular	O
representation	O
of	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
If	O
infinite	O
matrices	O
are	O
used	O
to	O
describe	O
linear	B-Architecture
maps	I-Architecture
,	O
then	O
only	O
those	O
matrices	O
can	O
be	O
used	O
all	O
of	O
whose	O
columns	O
have	O
but	O
a	O
finite	O
number	O
of	O
nonzero	O
entries	O
,	O
for	O
the	O
following	O
reason	O
.	O
</s>
<s>
For	O
a	O
matrix	O
A	O
to	O
describe	O
a	O
linear	B-Architecture
map	I-Architecture
f	O
:	O
V	O
→	O
W	O
,	O
bases	O
for	O
both	O
spaces	O
must	O
have	O
been	O
chosen	O
;	O
recall	O
that	O
by	O
definition	O
this	O
means	O
that	O
every	O
vector	O
in	O
the	O
space	O
can	O
be	O
written	O
uniquely	O
as	O
a	O
(	O
finite	O
)	O
linear	O
combination	O
of	O
basis	O
vectors	O
,	O
so	O
that	O
written	O
as	O
a	O
(	O
column	O
)	O
vectorv	O
of	O
coefficients	O
,	O
only	O
finitely	O
many	O
entries	O
v	O
are	O
nonzero	O
.	O
</s>
<s>
Products	O
of	O
two	O
matrices	O
of	O
the	O
given	O
type	O
are	O
well	O
defined	O
(	O
provided	O
that	O
the	O
column-index	O
and	O
row-index	O
sets	O
match	O
)	O
,	O
are	O
of	O
the	O
same	O
type	O
,	O
and	O
correspond	O
to	O
the	O
composition	B-Application
of	O
linear	B-Architecture
maps	I-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
the	O
matrices	O
whose	O
column	O
sums	O
are	O
absolutely	O
convergent	B-Algorithm
sequences	I-Algorithm
form	O
a	O
ring	O
.	O
</s>
<s>
However	O
,	O
the	O
explicit	O
point	O
of	O
view	O
of	O
matrices	O
tends	O
to	O
obfuscate	O
the	O
matter	O
,	O
and	O
the	O
abstract	O
and	O
more	O
powerful	O
tools	O
of	O
functional	B-Application
analysis	I-Application
can	O
be	O
used	O
instead	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
A	O
is	O
a	O
3-by-0	O
matrix	O
and	O
B	O
is	O
a	O
0-by-3	O
matrix	O
,	O
then	O
AB	O
is	O
the	O
3-by-3	O
zero	B-Algorithm
matrix	I-Algorithm
corresponding	O
to	O
the	O
null	O
map	O
from	O
a	O
3-dimensional	O
space	O
V	O
to	O
itself	O
,	O
while	O
BA	O
is	O
a	O
0-by-0	O
matrix	O
.	O
</s>
<s>
There	O
is	O
no	O
common	O
notation	O
for	O
empty	O
matrices	O
,	O
but	O
most	O
computer	B-General_Concept
algebra	I-General_Concept
systems	I-General_Concept
allow	O
creating	O
and	O
computing	O
with	O
them	O
.	O
</s>
<s>
Text	B-Algorithm
mining	I-Algorithm
and	O
automated	O
thesaurus	O
compilation	O
makes	O
use	O
of	O
document-term	B-General_Concept
matrices	I-General_Concept
such	O
as	O
tf-idf	O
to	O
track	O
frequencies	O
of	O
certain	O
words	O
in	O
several	O
documents	O
.	O
</s>
<s>
For	O
example	O
,	O
2-by-2	O
rotation	B-Algorithm
matrices	I-Algorithm
represent	O
the	O
multiplication	O
with	O
some	O
complex	O
number	O
of	O
absolute	O
value	O
1	O
,	O
as	O
above	O
.	O
</s>
<s>
Early	O
encryption	O
techniques	O
such	O
as	O
the	O
Hill	B-Algorithm
cipher	I-Algorithm
also	O
used	O
matrices	O
.	O
</s>
<s>
Computer	O
graphics	O
uses	O
matrices	O
to	O
represent	O
objects	O
;	O
to	O
calculate	O
transformations	O
of	O
objects	O
using	O
affine	O
rotation	B-Algorithm
matrices	I-Algorithm
to	O
accomplish	O
tasks	O
such	O
as	O
projecting	O
a	O
three-dimensional	O
object	O
onto	O
a	O
two-dimensional	O
screen	O
,	O
corresponding	O
to	O
a	O
theoretical	O
camera	O
observation	O
;	O
and	O
to	O
apply	O
image	O
convolutions	O
such	O
as	O
sharpening	O
,	O
blurring	O
,	O
edge	O
detection	O
,	O
and	O
more	O
.	O
</s>
<s>
The	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
finite	O
graph	O
is	O
a	O
basic	O
notion	O
of	O
graph	O
theory	O
.	O
</s>
<s>
Matrices	O
containing	O
just	O
two	O
different	O
values	O
(	O
1	O
and	O
0	O
meaning	O
for	O
example	O
"	O
yes	O
"	O
and	O
"	O
no	O
"	O
,	O
respectively	O
)	O
are	O
called	O
logical	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
These	O
concepts	O
can	O
be	O
applied	O
to	O
websites	O
connected	O
by	O
hyperlinks	O
or	O
cities	O
connected	O
by	O
roads	O
etc.	O
,	O
in	O
which	O
case	O
(	O
unless	O
the	O
connection	O
network	O
is	O
extremely	O
dense	O
)	O
the	O
matrices	O
tend	O
to	O
be	O
sparse	B-Algorithm
,	O
that	O
is	O
,	O
contain	O
few	O
nonzero	O
entries	O
.	O
</s>
<s>
The	O
Hessian	O
matrix	O
of	O
a	O
differentiable	O
function	O
ƒ	O
:	O
R	O
→	O
R	O
consists	O
of	O
the	O
second	B-Algorithm
derivatives	I-Algorithm
of	O
ƒ	O
with	O
respect	O
to	O
the	O
several	O
coordinate	O
directions	O
,	O
that	O
is	O
,	O
</s>
<s>
It	O
encodes	O
information	O
about	O
the	O
local	O
growth	O
behaviour	O
of	O
the	O
function	O
:	O
given	O
a	O
critical	O
point	O
x	O
=(	O
x	O
,...,	O
x	O
)	O
,	O
that	O
is	O
,	O
a	O
point	O
where	O
the	O
first	O
partial	O
derivatives	O
of	O
ƒ	O
vanish	O
,	O
the	O
function	O
has	O
a	O
local	O
minimum	O
if	O
the	O
Hessian	O
matrix	O
is	O
positive	B-Algorithm
definite	I-Algorithm
.	O
</s>
<s>
Quadratic	B-Algorithm
programming	I-Algorithm
can	O
be	O
used	O
to	O
find	O
global	O
minima	O
or	O
maxima	O
of	O
quadratic	O
functions	O
closely	O
related	O
to	O
the	O
ones	O
attached	O
to	O
matrices	O
(	O
see	O
above	O
)	O
.	O
</s>
<s>
For	O
elliptic	O
partial	O
differential	O
equations	O
this	O
matrix	O
is	O
positive	B-Algorithm
definite	I-Algorithm
,	O
which	O
has	O
a	O
decisive	O
influence	O
on	O
the	O
set	O
of	O
possible	O
solutions	O
of	O
the	O
equation	O
in	O
question	O
.	O
</s>
<s>
The	O
finite	B-Application
element	I-Application
method	I-Application
is	O
an	O
important	O
numerical	O
method	O
to	O
solve	O
partial	O
differential	O
equations	O
,	O
widely	O
applied	O
in	O
simulating	O
complex	O
physical	O
systems	O
.	O
</s>
<s>
It	O
attempts	O
to	O
approximate	O
the	O
solution	O
to	O
some	O
equation	O
by	O
piecewise	O
linear	O
functions	O
,	O
where	O
the	O
pieces	O
are	O
chosen	O
concerning	O
a	O
sufficiently	O
fine	O
grid	O
,	O
which	O
in	O
turn	O
can	O
be	O
recast	O
as	O
a	O
matrix	B-Architecture
equation	I-Architecture
.	O
</s>
<s>
Stochastic	B-Algorithm
matrices	I-Algorithm
are	O
square	B-Algorithm
matrices	I-Algorithm
whose	O
rows	O
are	O
probability	O
vectors	O
,	O
that	O
is	O
,	O
whose	O
entries	O
are	O
non-negative	O
and	O
sum	O
up	O
to	O
one	O
.	O
</s>
<s>
Stochastic	B-Algorithm
matrices	I-Algorithm
are	O
used	O
to	O
define	O
Markov	O
chains	O
with	O
finitely	O
many	O
states	O
.	O
</s>
<s>
A	O
row	O
of	O
the	O
stochastic	B-Algorithm
matrix	I-Algorithm
gives	O
the	O
probability	O
distribution	O
for	O
the	O
next	O
position	O
of	O
some	O
particle	O
currently	O
in	O
the	O
state	O
that	O
corresponds	O
to	O
the	O
row	O
.	O
</s>
<s>
Descriptive	B-General_Concept
statistics	I-General_Concept
is	O
concerned	O
with	O
describing	O
data	O
sets	O
,	O
which	O
can	O
often	O
be	O
represented	O
as	O
data	B-Algorithm
matrices	I-Algorithm
,	O
which	O
may	O
then	O
be	O
subjected	O
to	O
dimensionality	B-Algorithm
reduction	I-Algorithm
techniques	O
.	O
</s>
<s>
Random	O
matrices	O
are	O
matrices	O
whose	O
entries	O
are	O
random	O
numbers	O
,	O
subject	O
to	O
suitable	O
probability	O
distributions	O
,	O
such	O
as	O
matrix	O
normal	B-Algorithm
distribution	O
.	O
</s>
<s>
Linear	B-Architecture
transformations	I-Architecture
and	O
the	O
associated	O
symmetries	O
play	O
a	O
key	O
role	O
in	O
modern	O
physics	O
.	O
</s>
<s>
For	O
the	O
three	O
lightest	O
quarks	B-Operating_System
,	O
there	O
is	O
a	O
group-theoretical	O
representation	O
involving	O
the	O
special	O
unitary	B-Algorithm
group	O
SU(3 )	O
;	O
for	O
their	O
calculations	O
,	O
physicists	O
use	O
a	O
convenient	O
matrix	O
representation	O
known	O
as	O
the	O
Gell-Mann	O
matrices	O
,	O
which	O
are	O
also	O
used	O
for	O
the	O
SU(3 )	O
gauge	O
group	O
that	O
forms	O
the	O
basis	O
of	O
the	O
modern	O
description	O
of	O
strong	O
nuclear	O
interactions	O
,	O
quantum	O
chromodynamics	O
.	O
</s>
<s>
The	O
Cabibbo	O
–	O
Kobayashi	O
–	O
Maskawa	O
matrix	O
,	O
in	O
turn	O
,	O
expresses	O
the	O
fact	O
that	O
the	O
basic	O
quark	B-Operating_System
states	O
that	O
are	O
important	O
for	O
weak	O
interactions	O
are	O
not	O
the	O
same	O
as	O
,	O
but	O
linearly	O
related	O
to	O
the	O
basic	O
quark	B-Operating_System
states	O
that	O
define	O
particles	O
with	O
specific	O
and	O
distinct	O
masses	O
.	O
</s>
<s>
The	O
best	O
way	O
to	O
obtain	O
solutions	O
is	O
to	O
determine	O
the	O
system	O
's	O
eigenvectors	O
,	O
its	O
normal	B-Algorithm
modes	O
,	O
by	O
diagonalizing	B-Algorithm
the	O
matrix	B-Architecture
equation	I-Architecture
.	O
</s>
<s>
The	O
result	O
is	O
a	O
model	O
in	O
which	O
light	B-Device
rays	I-Device
are	O
indeed	O
geometrical	O
rays	O
.	O
</s>
<s>
If	O
the	O
deflection	O
of	O
light	B-Device
rays	I-Device
by	O
optical	O
elements	O
is	O
small	O
,	O
the	O
action	O
of	O
a	O
lens	O
or	O
reflective	O
element	O
on	O
a	O
given	O
light	B-Device
ray	I-Device
can	O
be	O
expressed	O
as	O
multiplication	O
of	O
a	O
two-component	O
vector	O
with	O
a	O
two-by-two	O
matrix	O
called	O
ray	O
transfer	O
matrix	O
analysis	O
:	O
the	O
vector	O
's	O
components	O
are	O
the	O
light	B-Device
ray	I-Device
's	O
slope	O
and	O
its	O
distance	O
from	O
the	O
optical	O
axis	O
,	O
while	O
the	O
matrix	O
encodes	O
the	O
properties	O
of	O
the	O
optical	O
element	O
.	O
</s>
<s>
Arthur	O
Cayley	O
published	O
a	O
treatise	O
on	O
geometric	B-Algorithm
transformations	I-Algorithm
using	O
matrices	O
that	O
were	O
not	O
rotated	O
versions	O
of	O
the	O
coefficients	O
being	O
investigated	O
as	O
had	O
previously	O
been	O
done	O
.	O
</s>
<s>
Early	O
matrix	B-Architecture
theory	I-Architecture
had	O
limited	O
the	O
use	O
of	O
arrays	O
almost	O
exclusively	O
to	O
determinants	O
and	O
Arthur	O
Cayley	O
's	O
abstract	O
matrix	O
operations	O
were	O
revolutionary	O
.	O
</s>
<s>
Number-theoretical	O
problems	O
led	O
Gauss	B-Algorithm
to	O
relate	O
coefficients	O
of	O
quadratic	O
forms	O
,	O
that	O
is	O
,	O
expressions	O
such	O
as	O
and	O
linear	B-Architecture
maps	I-Architecture
in	O
three	O
dimensions	O
to	O
matrices	O
.	O
</s>
<s>
He	O
also	O
showed	O
,	O
in	O
1829	O
,	O
that	O
the	O
eigenvalues	O
of	O
symmetric	B-Algorithm
matrices	I-Algorithm
are	O
real	O
.	O
</s>
<s>
Jacobi	O
studied	O
"	O
functional	O
determinants	O
"	O
—	O
later	O
called	O
Jacobi	O
determinants	O
by	O
Sylvester	O
—	O
which	O
can	O
be	O
used	O
to	O
describe	O
geometric	B-Algorithm
transformations	I-Algorithm
at	O
a	O
local	O
(	O
or	O
infinitesimal	O
)	O
level	O
,	O
see	O
above	O
;	O
Kronecker	O
's	O
Vorlesungen	O
über	O
die	O
Theorie	O
der	O
Determinanten	O
and	O
Weierstrass	O
 '	O
Zur	O
Determinantentheorie	O
,	O
both	O
published	O
in	O
1903	O
,	O
first	O
treated	O
determinants	O
axiomatically	O
,	O
as	O
opposed	O
to	O
previous	O
more	O
concrete	O
approaches	O
such	O
as	O
the	O
mentioned	O
formula	O
of	O
Cauchy	O
.	O
</s>
<s>
Also	O
at	O
the	O
end	O
of	O
the	O
19th	O
century	O
,	O
the	O
Gauss	B-Algorithm
–	I-Algorithm
Jordan	I-Algorithm
elimination	I-Algorithm
(	O
generalizing	O
a	O
special	O
case	O
now	O
known	O
as	O
Gauss	B-Algorithm
elimination	I-Algorithm
)	O
was	O
established	O
by	O
Wilhelm	O
Jordan	O
.	O
</s>
<s>
In	O
the	O
early	O
20th	O
century	O
,	O
matrices	O
attained	O
a	O
central	O
role	O
in	O
linear	B-Language
algebra	I-Language
,	O
partially	O
due	O
to	O
their	O
use	O
in	O
classification	O
of	O
the	O
hypercomplex	O
number	O
systems	O
of	O
the	O
previous	O
century	O
.	O
</s>
<s>
Later	O
,	O
von	O
Neumann	O
carried	O
out	O
the	O
mathematical	O
formulation	O
of	O
quantum	O
mechanics	O
,	O
by	O
further	O
developing	O
functional	B-Application
analytic	I-Application
notions	O
such	O
as	O
linear	B-Architecture
operators	I-Architecture
on	O
Hilbert	O
spaces	O
,	O
which	O
,	O
very	O
roughly	O
speaking	O
,	O
correspond	O
to	O
Euclidean	O
space	O
,	O
but	O
with	O
an	O
infinity	B-Application
of	O
independent	O
directions	O
.	O
</s>
<s>
Bertrand	O
Russell	O
and	O
Alfred	O
North	O
Whitehead	O
in	O
their	O
Principia	O
Mathematica	B-Language
(	O
1910	O
–	O
1913	O
)	O
use	O
the	O
word	O
"	O
matrix	O
"	O
in	O
the	O
context	O
of	O
their	O
axiom	B-Algorithm
of	O
reducibility	O
.	O
</s>
<s>
They	O
proposed	O
this	O
axiom	B-Algorithm
as	O
a	O
means	O
to	O
reduce	O
any	O
function	O
to	O
one	O
of	O
lower	O
type	O
,	O
successively	O
,	O
so	O
that	O
at	O
the	O
"	O
bottom	O
"	O
(	O
0	O
order	O
)	O
the	O
function	O
is	O
identical	O
to	O
its	O
extension	O
:	O
</s>
