<s>
In	O
mathematics	O
,	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
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	B-Architecture
is	O
known	O
as	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
order	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>
Square	B-Algorithm
matrices	I-Algorithm
are	O
often	O
used	O
to	O
represent	O
simple	O
linear	B-Architecture
transformations	I-Architecture
,	O
such	O
as	O
shearing	B-Algorithm
or	O
rotation	B-General_Concept
.	O
</s>
<s>
For	O
example	O
,	O
if	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
representing	O
a	O
rotation	B-General_Concept
(	O
rotation	B-Algorithm
matrix	I-Algorithm
)	O
and	O
is	O
a	O
column	O
vector	O
describing	O
the	O
position	O
of	O
a	O
point	O
in	O
space	O
,	O
the	O
product	O
yields	O
another	O
column	O
vector	O
describing	O
the	O
position	O
of	O
that	O
point	O
after	O
that	O
rotation	B-General_Concept
.	O
</s>
<s>
The	O
entries	O
(	O
i	O
=	O
1	O
,	O
…,	O
n	O
)	O
form	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
They	O
lie	O
on	O
the	O
imaginary	O
line	O
which	O
runs	O
from	O
the	O
top	O
left	O
corner	O
to	O
the	O
bottom	O
right	O
corner	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
the	O
4×4	O
matrix	B-Architecture
above	O
contains	O
the	O
elements	O
,	O
,	O
,	O
.	O
</s>
<s>
The	O
diagonal	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
from	O
the	O
top	O
right	O
to	O
the	O
bottom	O
left	O
corner	O
is	O
called	O
antidiagonal	B-Algorithm
or	O
counterdiagonal	B-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
is	O
called	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
only	O
all	O
entries	O
above	O
(	O
or	O
below	O
)	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
are	O
zero	O
,	O
is	O
called	O
an	O
upper	O
(	O
or	O
lower	O
)	O
triangular	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
size	O
is	O
the	O
matrix	B-Architecture
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
e.g.	O
</s>
<s>
It	O
is	O
a	O
square	B-Algorithm
matrix	I-Algorithm
of	O
order	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
identity	B-Algorithm
matrix	I-Algorithm
because	O
multiplication	O
with	O
it	O
leaves	O
a	O
matrix	B-Architecture
unchanged	O
:	O
</s>
<s>
A	O
square	B-Algorithm
matrix	I-Algorithm
that	O
is	O
equal	O
to	O
its	O
transpose	O
,	O
i.e.	O
,	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
If	O
instead	O
then	O
is	O
called	O
a	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
a	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
often	O
the	O
appropriate	O
analogue	O
of	O
the	O
transpose	O
is	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
defined	O
as	O
the	O
transpose	O
of	O
the	O
complex	O
conjugate	O
of	O
A	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
satisfying	O
is	O
called	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
By	O
the	O
spectral	O
theorem	O
,	O
real	O
symmetric	B-Algorithm
(	O
or	O
complex	O
Hermitian	O
)	O
matrices	O
have	O
an	O
orthogonal	O
(	O
or	O
unitary	B-Algorithm
)	O
eigenbasis	O
;	O
i.e.	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>
Positive	B-Algorithm
definite	I-Algorithm
Indefinite	B-Algorithm
Q(x,y )	O
=	O
1/4	O
x2	O
+	O
y2	O
Q(x,y )	O
=	O
1/4	O
x2	O
−	O
1/4	O
y2	O
150px	O
Points	O
such	O
that	O
(	O
Ellipse	O
)	O
.	O
</s>
<s>
If	O
the	O
quadratic	O
form	O
takes	O
only	O
non-negative	O
(	O
respectively	O
only	O
non-positive	O
)	O
values	O
,	O
the	O
symmetric	B-Algorithm
matrix	I-Algorithm
is	O
called	O
positive-semidefinite	O
(	O
respectively	O
negative-semidefinite	O
)	O
;	O
hence	O
the	O
matrix	B-Architecture
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	B-Algorithm
if	O
and	O
only	O
if	O
all	O
its	O
eigenvalues	O
are	O
positive	O
.	O
</s>
<s>
An	O
orthogonal	O
matrix	B-Architecture
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
i.e.	O
,	O
orthonormal	B-Algorithm
vectors	I-Algorithm
)	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
matrix	B-Architecture
A	O
is	O
orthogonal	O
if	O
its	O
transpose	O
is	O
equal	O
to	O
its	O
inverse	O
:	O
</s>
<s>
where	O
I	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
orthogonal	O
matrix	B-Architecture
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	O
matrix	B-Architecture
is	O
either	O
+1	O
or	O
−1	O
.	O
</s>
<s>
The	O
complex	O
analogue	O
of	O
an	O
orthogonal	O
matrix	B-Architecture
is	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
real	O
or	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
is	O
called	O
normal	B-Algorithm
if	O
If	O
a	O
real	B-Algorithm
square	I-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
,	O
skew-symmetric	O
,	O
or	O
orthogonal	O
,	O
then	O
it	O
is	O
normal	B-Algorithm
.	O
</s>
<s>
If	O
a	O
complex	O
square	B-Algorithm
matrix	I-Algorithm
is	O
Hermitian	O
,	O
skew-Hermitian	O
,	O
or	O
unitary	B-Algorithm
,	O
then	O
it	O
is	O
normal	B-Algorithm
.	O
</s>
<s>
Normal	B-Algorithm
matrices	I-Algorithm
are	O
of	O
interest	O
mainly	O
because	O
they	O
include	O
the	O
types	O
of	O
matrices	O
just	O
listed	O
and	O
form	O
the	O
broadest	O
class	O
of	O
matrices	O
for	O
which	O
the	O
spectral	O
theorem	O
holds	O
.	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>
While	O
matrix	B-Architecture
multiplication	O
is	O
not	O
commutative	O
,	O
the	O
trace	O
of	O
the	O
product	O
of	O
two	O
matrices	O
is	O
independent	O
of	O
the	O
order	O
of	O
the	O
factors	O
:	O
</s>
<s>
This	O
is	O
immediate	O
from	O
the	O
definition	O
of	O
matrix	B-Architecture
multiplication	O
:	O
</s>
<s>
Also	O
,	O
the	O
trace	O
of	O
a	O
matrix	B-Architecture
is	O
equal	O
to	O
that	O
of	O
its	O
transpose	O
,	O
i.e.	O
,	O
</s>
<s>
The	O
determinant	O
or	O
of	O
a	O
square	B-Algorithm
matrix	I-Algorithm
is	O
a	O
number	O
encoding	O
certain	O
properties	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
A	O
matrix	B-Architecture
is	O
invertible	O
if	O
and	O
only	O
if	O
its	O
determinant	O
is	O
nonzero	O
.	O
</s>
<s>
Its	O
absolute	O
value	O
equals	O
the	O
area	O
(	O
in	O
)	O
or	O
volume	O
(	O
in	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	B-Architecture
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
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	B-Architecture
.	O
</s>
<s>
This	O
expansion	O
can	O
be	O
used	O
for	O
a	O
recursive	O
definition	O
of	O
determinants	O
(	O
taking	O
as	O
starting	O
case	O
the	O
determinant	O
of	O
a	O
1×1	O
matrix	B-Architecture
,	O
which	O
is	O
its	O
unique	O
entry	O
,	O
or	O
even	O
the	O
determinant	O
of	O
a	O
0×0	O
matrix	B-Architecture
,	O
which	O
is	O
1	O
)	O
,	O
that	O
can	O
be	O
seen	O
to	O
be	O
equivalent	O
to	O
the	O
Leibniz	O
formula	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>
It	O
is	O
a	O
monic	O
polynomial	O
of	O
degree	O
n	O
.	O
Therefore	O
the	O
polynomial	O
equation	O
has	O
at	O
most	O
n	O
different	O
solutions	O
,	O
i.e.	O
,	O
eigenvalues	O
of	O
the	O
matrix	B-Architecture
.	O
</s>
<s>
According	O
to	O
the	O
Cayley	O
–	O
Hamilton	O
theorem	O
,	O
,	O
that	O
is	O
,	O
the	O
result	O
of	O
substituting	O
the	O
matrix	B-Architecture
itself	O
into	O
its	O
own	O
characteristic	O
polynomial	O
yields	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
