<s>
In	O
mathematics	O
,	O
particularly	O
in	O
linear	B-Language
algebra	I-Language
and	O
applications	O
,	O
matrix	B-Algorithm
analysis	I-Algorithm
is	O
the	O
study	O
of	O
matrices	B-Architecture
and	O
their	O
algebraic	O
properties	O
.	O
</s>
<s>
Some	O
particular	O
topics	O
out	O
of	O
many	O
include	O
;	O
operations	O
defined	O
on	O
matrices	B-Architecture
(	O
such	O
as	O
matrix	O
addition	O
,	O
matrix	O
multiplication	O
and	O
operations	O
derived	O
from	O
these	O
)	O
,	O
functions	O
of	O
matrices	B-Architecture
(	O
such	O
as	O
matrix	O
exponentiation	O
and	O
matrix	O
logarithm	O
,	O
and	O
even	O
sines	O
and	O
cosines	O
etc	O
.	O
</s>
<s>
of	O
matrices	B-Architecture
)	O
,	O
and	O
the	O
eigenvalues	O
of	O
matrices	B-Architecture
(	O
eigendecomposition	O
of	O
a	O
matrix	O
,	O
eigenvalue	O
perturbation	O
theory	O
)	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
m×n	O
matrices	B-Architecture
over	O
a	O
field	O
F	O
denoted	O
in	O
this	O
article	O
Mmn(F )	O
form	O
a	O
vector	O
space	O
.	O
</s>
<s>
Two	O
m×n	O
matrices	B-Architecture
A	O
and	O
B	O
in	O
Mmn(F )	O
can	O
be	O
added	O
together	O
to	O
form	O
another	O
matrix	O
in	O
the	O
space	O
Mmn(F )	O
:	O
</s>
<s>
Combining	O
these	O
two	O
properties	O
,	O
a	O
linear	O
combination	O
of	O
matrices	B-Architecture
A	O
and	O
B	O
are	O
in	O
Mmn(F )	O
is	O
another	O
matrix	O
in	O
Mmn(F )	O
:	O
</s>
<s>
Any	O
matrix	O
can	O
be	O
expressed	O
as	O
a	O
linear	O
combination	O
of	O
basis	O
matrices	B-Architecture
,	O
which	O
play	O
the	O
role	O
of	O
the	O
basis	O
vectors	O
for	O
the	O
matrix	O
space	O
.	O
</s>
<s>
For	O
example	O
,	O
for	O
the	O
set	O
of	O
2×2	O
matrices	B-Architecture
over	O
the	O
field	O
of	O
real	O
numbers	O
,	O
,	O
one	O
legitimate	O
basis	O
set	O
of	O
matrices	B-Architecture
is	O
:	O
</s>
<s>
This	O
idea	O
applies	O
to	O
other	O
fields	O
and	O
matrices	B-Architecture
of	O
higher	O
dimensions	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
an	O
important	O
property	O
.	O
</s>
<s>
Determinants	O
are	O
used	O
for	O
finding	O
eigenvalues	O
of	O
matrices	B-Architecture
(	O
see	O
below	O
)	O
,	O
and	O
for	O
solving	O
a	O
system	O
of	O
linear	O
equations	O
(	O
see	O
Cramer	O
's	O
rule	O
)	O
.	O
</s>
<s>
where	O
I	O
is	O
the	O
n×n	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Two	O
n×n	O
matrices	B-Architecture
A	O
and	O
B	O
are	O
similar	O
if	O
they	O
are	O
related	O
by	O
a	O
similarity	O
transformation	O
:	O
</s>
<s>
LU	O
decomposition	O
splits	O
a	O
matrix	O
into	O
a	O
matrix	O
product	O
of	O
an	O
upper	B-Algorithm
triangular	I-Algorithm
matrix	I-Algorithm
and	O
a	O
lower	O
triangle	O
matrix	O
.	O
</s>
<s>
Since	O
matrices	B-Architecture
form	O
vector	O
spaces	O
,	O
one	O
can	O
form	O
axioms	O
(	O
analogous	O
to	O
those	O
of	O
vectors	O
)	O
to	O
define	O
a	O
"	O
size	O
"	O
of	O
a	O
particular	O
matrix	O
.	O
</s>
<s>
For	O
all	O
matrices	B-Architecture
A	O
and	O
B	O
in	O
Mmn(F )	O
,	O
and	O
all	O
numbers	O
α	O
in	O
F	O
,	O
a	O
matrix	O
norm	O
,	O
delimited	O
by	O
double	O
vertical	O
bars	O
||	O
...	O
||	O
,	O
fulfills	O
:	O
</s>
<s>
with	O
equality	O
only	O
for	O
A	O
=	O
0	O
,	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
is	O
defined	O
for	O
matrices	B-Architecture
of	O
any	O
dimension	O
(	O
i.e.	O
</s>
<s>
no	O
restriction	O
to	O
square	B-Algorithm
matrices	I-Algorithm
)	O
.	O
</s>
