<s>
This	O
article	O
lists	O
some	O
important	O
classes	O
of	O
matrices	B-Architecture
used	O
in	O
mathematics	O
,	O
science	O
and	O
engineering	O
.	O
</s>
<s>
A	O
matrix	O
(	O
plural	O
matrices	B-Architecture
,	O
or	O
less	O
commonly	O
matrixes	O
)	O
is	O
a	O
rectangular	O
array	B-Data_Structure
of	O
numbers	O
called	O
entries	O
.	O
</s>
<s>
Matrices	B-Architecture
have	O
a	O
long	O
history	O
of	O
both	O
study	O
and	O
application	O
,	O
leading	O
to	O
diverse	O
ways	O
of	O
classifying	O
matrices	B-Architecture
.	O
</s>
<s>
A	O
first	O
group	O
is	O
matrices	B-Architecture
satisfying	O
concrete	O
conditions	O
of	O
the	O
entries	O
,	O
including	O
constant	O
matrices	B-Architecture
.	O
</s>
<s>
and	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
of	O
dimension	O
.	O
</s>
<s>
Further	O
ways	O
of	O
classifying	O
matrices	B-Architecture
are	O
according	O
to	O
their	O
eigenvalues	O
,	O
or	O
by	O
imposing	O
conditions	O
on	O
the	O
product	O
of	O
the	O
matrix	O
with	O
other	O
matrices	B-Architecture
.	O
</s>
<s>
Finally	O
,	O
many	O
domains	O
,	O
both	O
in	O
mathematics	O
and	O
other	O
sciences	O
including	O
physics	O
and	O
chemistry	O
,	O
have	O
particular	O
matrices	B-Architecture
that	O
are	O
applied	O
chiefly	O
in	O
these	O
areas	O
.	O
</s>
<s>
The	O
list	O
below	O
comprises	O
matrices	B-Architecture
whose	O
elements	O
are	O
constant	O
for	O
any	O
given	O
dimension	O
(	O
size	O
)	O
of	O
matrix	O
.	O
</s>
<s>
Name	O
Explanation	O
Symbolic	O
description	O
of	O
the	O
entries	O
Notes	O
Commutation	B-Algorithm
matrix	I-Algorithm
The	O
matrix	O
of	O
the	O
linear	B-Architecture
map	I-Architecture
that	O
maps	O
a	O
matrix	O
to	O
its	O
transpose	O
See	O
Vectorization	B-Algorithm
Duplication	B-Algorithm
matrix	I-Algorithm
The	O
matrix	O
of	O
the	O
linear	B-Architecture
map	I-Architecture
mapping	O
the	O
vector	O
of	O
the	O
distinct	O
entries	O
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
to	O
the	O
vector	O
of	O
all	O
enties	O
of	O
the	O
matrix	O
See	O
Vectorization	B-Algorithm
Elimination	B-Algorithm
matrix	I-Algorithm
The	O
matrix	O
of	O
the	O
linear	B-Architecture
map	I-Architecture
mapping	O
the	O
vector	O
of	O
the	O
entries	O
of	O
a	O
matrix	O
to	O
the	O
vector	O
of	O
a	O
part	O
of	O
the	O
entries	O
(	O
for	O
example	O
the	O
vector	O
of	O
the	O
entries	O
that	O
are	O
not	O
below	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
)	O
See	O
vectorization	B-Algorithm
Exchange	O
matrix	O
The	O
binary	B-Algorithm
matrix	I-Algorithm
with	O
ones	O
on	O
the	O
anti-diagonal	B-Algorithm
,	O
and	O
zeroes	O
everywhere	O
else	O
.	O
</s>
<s>
aij	O
=	O
δn+1−i	O
,	O
j	O
A	O
permutation	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
Hankel	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Identity	B-Algorithm
matrix	I-Algorithm
A	O
square	O
diagonal	B-Algorithm
matrix	I-Algorithm
,	O
with	O
all	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
equal	O
to	O
1	O
,	O
and	O
the	O
rest	O
0	O
.	O
aij	O
=	O
δij	O
Lehmer	B-Algorithm
matrix	I-Algorithm
aij	O
=	O
min(i, j )	O
÷	O
max(i, j )	O
.	O
</s>
<s>
A	O
positive	B-Algorithm
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Pascal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
containing	O
the	O
entries	O
of	O
Pascal	O
's	O
triangle	O
.	O
</s>
<s>
Pauli	O
matrices	B-Architecture
A	O
set	O
of	O
three	O
2	O
×	O
2	O
complex	O
Hermitian	B-Algorithm
and	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
When	O
combined	O
with	O
the	O
I2	O
identity	B-Algorithm
matrix	I-Algorithm
,	O
they	O
form	O
an	O
orthogonal	O
basis	O
for	O
the	O
2	O
×	O
2	O
complex	O
Hermitian	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Shift	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
ones	O
on	O
the	O
superdiagonal	O
or	O
subdiagonal	O
and	O
zeroes	O
elsewhere	O
.	O
</s>
<s>
Zero	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
entries	O
equal	O
to	O
zero	O
.	O
</s>
<s>
The	O
following	O
lists	O
matrices	B-Architecture
whose	O
entries	O
are	O
subject	O
to	O
certain	O
conditions	O
.	O
</s>
<s>
Many	O
of	O
them	O
apply	O
to	O
square	O
matrices	B-Architecture
only	O
,	O
that	O
is	O
matrices	B-Architecture
with	O
the	O
same	O
number	O
of	O
columns	O
and	O
rows	O
.	O
</s>
<s>
The	O
main	B-Algorithm
diagonal	I-Algorithm
of	O
a	O
square	O
matrix	O
is	O
the	O
diagonal	O
joining	O
the	O
upper	O
left	O
corner	O
and	O
the	O
lower	O
right	O
one	O
or	O
equivalently	O
the	O
entries	O
ai	B-Application
,	O
i	O
.	O
</s>
<s>
The	O
other	O
diagonal	O
is	O
called	O
anti-diagonal	B-Algorithm
(	O
or	O
counter-diagonal	O
)	O
.	O
</s>
<s>
Synonym	O
for	O
binary	B-Algorithm
matrix	I-Algorithm
or	O
logical	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Alternant	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
in	O
which	O
successive	O
columns	O
have	O
a	O
particular	O
function	O
applied	O
to	O
their	O
entries	O
.	O
</s>
<s>
Alternating	B-Algorithm
sign	I-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
entries	O
0	O
,	O
1	O
and	O
−1	O
such	O
that	O
the	O
sum	O
of	O
each	O
row	O
and	O
column	O
is	O
1	O
and	O
the	O
nonzero	O
entries	O
in	O
each	O
row	O
and	O
column	O
alternate	O
in	O
sign	O
.	O
</s>
<s>
Anti-diagonal	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
all	O
entries	O
off	O
the	O
anti-diagonal	B-Algorithm
equal	O
to	O
zero	O
.	O
</s>
<s>
Anti-symmetric	B-Algorithm
matrix	I-Algorithm
Synonym	O
for	O
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Arrowhead	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
containing	O
zeros	O
in	O
all	O
entries	O
except	O
for	O
the	O
first	O
row	O
,	O
first	O
column	O
,	O
and	O
main	B-Algorithm
diagonal	I-Algorithm
.	O
</s>
<s>
Band	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
whose	O
non-zero	O
entries	O
are	O
confined	O
to	O
a	O
diagonal	O
band	O
.	O
</s>
<s>
Bidiagonal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
elements	O
only	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
either	O
the	O
superdiagonal	O
or	O
subdiagonal	O
.	O
</s>
<s>
Binary	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
all	O
either	O
0	O
or	O
1	O
.	O
</s>
<s>
Synonym	O
for	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
or	O
logical	B-Algorithm
matrix	I-Algorithm
..	O
Bisymmetric	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
that	O
is	O
symmetric	B-Algorithm
with	O
respect	O
to	O
its	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
its	O
main	O
cross-diagonal	O
.	O
</s>
<s>
Block-diagonal	O
matrix	O
A	O
block	B-Algorithm
matrix	I-Algorithm
with	O
entries	O
only	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Block	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
partitioned	O
in	O
sub-matrices	O
called	O
blocks	O
.	O
</s>
<s>
Block	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
A	O
block	B-Algorithm
matrix	I-Algorithm
which	O
is	O
essentially	O
a	O
tridiagonal	B-Algorithm
matrix	I-Algorithm
but	O
with	O
submatrices	O
in	O
place	O
of	O
scalar	O
elements	O
.	O
</s>
<s>
Boolean	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
taken	O
from	O
a	O
Boolean	O
algebra	O
.	O
</s>
<s>
Cauchy	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
elements	O
are	O
of	O
the	O
form	O
1/	O
( xi	O
+	O
yj	O
)	O
for	O
(	O
xi	O
)	O
,	O
(	O
yj	O
)	O
injective	O
sequences	O
(	O
i.e.	O
,	O
taking	O
every	O
value	O
only	O
once	O
)	O
.	O
</s>
<s>
Centrosymmetric	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
symmetric	B-Algorithm
about	O
its	O
center	O
;	O
i.e.	O
,	O
aij	O
=	O
an−i+1	O
,	O
n−j+1	O
.	O
</s>
<s>
Conference	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
zero	O
diagonal	O
and	O
+1	O
and	O
−1	O
off	O
the	O
diagonal	O
,	O
such	O
that	O
CTC	O
is	O
a	O
multiple	O
of	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Complex	B-Algorithm
Hadamard	I-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
rows	O
and	O
columns	O
mutually	O
orthogonal	O
,	O
whose	O
entries	O
are	O
unimodular	B-Algorithm
.	O
</s>
<s>
Compound	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
generated	O
by	O
the	O
determinants	O
of	O
all	O
minors	O
of	O
a	O
matrix	O
.	O
</s>
<s>
Copositive	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
A	O
with	O
real	O
coefficients	O
,	O
such	O
that	O
is	O
nonnegative	O
for	O
every	O
nonnegative	O
vector	O
x	O
Diagonally	B-Algorithm
dominant	I-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
satisfy	O
.	O
</s>
<s>
Diagonal	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
all	O
entries	O
outside	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
equal	O
to	O
zero	O
.	O
</s>
<s>
Elementary	O
matrix	O
A	O
square	O
matrix	O
derived	O
by	O
applying	O
an	O
elementary	O
row	O
operation	O
to	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Equivalent	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
that	O
can	O
be	O
derived	O
from	O
another	O
matrix	O
through	O
a	O
sequence	O
of	O
elementary	O
row	O
or	O
column	O
operations	O
.	O
</s>
<s>
Frobenius	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
in	O
the	O
form	O
of	O
an	O
identity	B-Algorithm
matrix	I-Algorithm
but	O
with	O
arbitrary	O
entries	O
in	O
one	O
column	O
below	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
.	O
</s>
<s>
Generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
precisely	O
one	O
nonzero	O
element	O
in	O
each	O
row	O
and	O
column	O
.	O
</s>
<s>
Hankel	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
constant	O
skew-diagonals	O
;	O
also	O
an	O
upside	O
down	O
Toeplitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
square	O
Hankel	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
Hermitian	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
which	O
is	O
equal	O
to	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
A	O
=	O
A*	O
.	O
</s>
<s>
Hessenberg	B-Algorithm
matrix	I-Algorithm
An	O
"	O
almost	O
"	O
triangular	B-Algorithm
matrix	I-Algorithm
,	O
for	O
example	O
,	O
an	O
upper	B-Algorithm
Hessenberg	I-Algorithm
matrix	I-Algorithm
has	O
zero	O
entries	O
below	O
the	O
first	O
subdiagonal	O
.	O
</s>
<s>
Hollow	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
whose	O
main	B-Algorithm
diagonal	I-Algorithm
comprises	O
only	O
zero	O
elements	O
.	O
</s>
<s>
Integer	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
all	O
integers	O
.	O
</s>
<s>
Logical	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
entries	O
either	O
0	O
or	O
1	O
.	O
</s>
<s>
Synonym	O
for	O
(	O
0	O
,	O
1	O
)	O
-matrix	O
,	O
binary	B-Algorithm
matrix	I-Algorithm
or	O
Boolean	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Can	O
be	O
used	O
to	O
represent	O
a	O
k-adic	O
relation	B-Algorithm
.	O
</s>
<s>
Markov	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
of	O
non-negative	B-Algorithm
real	O
numbers	O
,	O
such	O
that	O
the	O
entries	O
in	O
each	O
row	O
sum	O
to	O
1	O
.	O
</s>
<s>
Metzler	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
off-diagonal	O
entries	O
are	O
non-negative	B-Algorithm
.	O
</s>
<s>
Monomial	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
with	O
exactly	O
one	O
non-zero	O
entry	O
in	O
each	O
row	O
and	O
column	O
.	O
</s>
<s>
Synonym	O
for	O
generalized	B-Algorithm
permutation	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Nonnegative	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
nonnegative	O
entries	O
.	O
</s>
<s>
Null-symmetric	O
matrixA	O
square	O
matrix	O
whose	O
null	B-Algorithm
space	I-Algorithm
(	O
or	O
kernel	B-Algorithm
)	O
is	O
equal	O
to	O
its	O
transpose	O
,	O
N(A )	O
=	O
N(AT )	O
or	O
ker(A )	O
=	O
ker(AT )	O
.Synonym	O
for	O
kernel-symmetric	O
matrices	O
.	O
</s>
<s>
Examples	O
include	O
(	O
but	O
not	O
limited	O
to	O
)	O
symmetric	B-Algorithm
,	O
skew-symmetric	O
,	O
and	O
normal	O
matrices.Null-Hermitian	O
matrixA	O
square	O
matrix	O
whose	O
null	B-Algorithm
space	I-Algorithm
(	O
or	O
kernel	B-Algorithm
)	O
is	O
equal	O
to	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
N(A )	O
=	O
N( A*	O
)	O
or	O
ker(A )	O
=	O
ker( A*	O
)	O
.Synonym	O
for	O
kernel-Hermitian	O
matrices	O
.	O
</s>
<s>
Examples	O
include	O
(	O
but	O
not	O
limited	O
)	O
to	O
Hermitian	B-Algorithm
,	O
skew-Hermitian	O
matrices	O
,	O
and	O
normal	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Partitioned	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
partitioned	O
into	O
sub-matrices	O
,	O
or	O
equivalently	O
,	O
a	O
matrix	O
whose	O
entries	O
are	O
themselves	O
matrices	B-Architecture
rather	O
than	O
scalars	O
.	O
</s>
<s>
Synonym	O
for	O
block	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Pentadiagonal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
the	O
only	O
nonzero	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
the	O
two	O
diagonals	O
just	O
above	O
and	O
below	O
the	O
main	O
one	O
.	O
</s>
<s>
Permutation	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
representation	O
of	O
a	O
permutation	B-Algorithm
,	O
a	O
square	O
matrix	O
with	O
exactly	O
one	O
1	O
in	O
each	O
row	O
and	O
column	O
,	O
and	O
all	O
other	O
elements	O
0	O
.	O
</s>
<s>
Persymmetric	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
that	O
is	O
symmetric	B-Algorithm
about	O
its	O
northeast	O
–	O
southwest	O
diagonal	O
,	O
i.e.	O
,	O
aij	O
=	O
an−j+1	O
,	O
n−i+1	O
.	O
</s>
<s>
Positive	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
positive	B-Algorithm
entries	O
.	O
</s>
<s>
Quaternionic	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
quaternions	O
.	O
</s>
<s>
Random	O
matrix	O
A	O
matrix	O
whose	O
entries	O
are	O
random	O
variables	O
Sign	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
entries	O
are	O
either	O
+1	O
,	O
0	O
,	O
or	O
−1	O
.	O
</s>
<s>
Signature	B-Algorithm
matrix	I-Algorithm
A	O
diagonal	B-Algorithm
matrix	I-Algorithm
where	O
the	O
diagonal	O
elements	O
are	O
either	O
+1	O
or	O
−1	O
.	O
</s>
<s>
Skew-Hermitian	O
matrix	O
A	O
square	O
matrix	O
which	O
is	O
equal	O
to	O
the	O
negative	O
of	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
A*	O
=	O
−A	O
.	O
</s>
<s>
Skew-symmetric	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
which	O
is	O
equal	O
to	O
the	O
negative	O
of	O
its	O
transpose	O
,	O
AT	O
=	O
−A	O
.	O
</s>
<s>
Skyline	B-Algorithm
matrix	I-Algorithm
A	O
rearrangement	O
of	O
the	O
entries	O
of	O
a	O
banded	B-Algorithm
matrix	I-Algorithm
which	O
requires	O
less	O
space	O
.	O
</s>
<s>
Sparse	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
relatively	O
few	O
non-zero	O
elements	O
.	O
</s>
<s>
Sparse	B-Algorithm
matrix	I-Algorithm
algorithms	O
can	O
tackle	O
huge	O
sparse	B-Algorithm
matrices	I-Algorithm
that	O
are	O
utterly	O
impractical	O
for	O
dense	B-Algorithm
matrix	I-Algorithm
algorithms	O
.	O
</s>
<s>
Symmetric	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
which	O
is	O
equal	O
to	O
its	O
transpose	O
,	O
A	O
=	O
AT	O
(	O
ai	B-Application
,	O
j	O
=	O
aj	O
,	O
i	O
)	O
.	O
</s>
<s>
Toeplitz	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
constant	O
diagonals	O
.	O
</s>
<s>
Totally	O
positive	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
determinants	O
of	O
all	O
its	O
square	O
submatrices	O
positive	B-Algorithm
.	O
</s>
<s>
Triangular	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
all	O
entries	O
above	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
equal	O
to	O
zero	O
(	O
lower	B-Algorithm
triangular	I-Algorithm
)	O
or	O
with	O
all	O
entries	O
below	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
equal	O
to	O
zero	O
(	O
upper	B-Algorithm
triangular	I-Algorithm
)	O
.	O
</s>
<s>
Tridiagonal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
with	O
the	O
only	O
nonzero	O
entries	O
on	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
and	O
the	O
diagonals	O
just	O
above	O
and	O
below	O
the	O
main	O
one.X	O
–	O
Y	O
–	O
Z	O
matrixA	O
generalization	O
to	O
three	O
dimensions	O
of	O
the	O
concept	O
of	O
two-dimensional	O
array	B-Data_Structure
Vandermonde	O
matrix	O
A	O
row	O
consists	O
of	O
1	O
,	O
a	O
,	O
a2	O
,	O
a3	O
,	O
etc.	O
,	O
and	O
each	O
row	O
uses	O
a	O
different	O
variable	O
.	O
</s>
<s>
Walsh	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
,	O
with	O
dimensions	O
a	O
power	O
of	O
2	O
,	O
the	O
entries	O
of	O
which	O
are	O
+1	O
or	O
−1	O
,	O
and	O
the	O
property	O
that	O
the	O
dot	O
product	O
of	O
any	O
two	O
distinct	O
rows	O
(	O
or	O
columns	O
)	O
is	O
zero	O
.	O
</s>
<s>
Z-matrix	B-Algorithm
A	O
matrix	O
with	O
all	O
off-diagonal	O
entries	O
less	O
than	O
zero	O
.	O
</s>
<s>
An	O
inverse	O
of	O
square	O
matrix	O
A	B-Application
is	I-Application
a	O
matrix	O
B	O
(	O
necessarily	O
of	O
the	O
same	O
dimension	O
as	O
A	O
)	O
such	O
that	O
AB	O
=	O
I	O
.	O
Equivalently	O
,	O
BA	O
=	O
I	O
.	O
</s>
<s>
Name	O
Explanation	O
NotesCircular	O
matrix	O
or	O
Coninvolutory	O
matrixA	O
matrix	O
whose	O
inverse	O
is	O
equal	O
to	O
its	O
entrywise	O
complex	O
conjugate	O
:	O
A−1	O
=	O
.Compare	O
with	O
unitary	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Congruent	O
matrix	O
Two	O
matrices	B-Architecture
A	O
and	O
B	O
are	O
congruent	O
if	O
there	O
exists	O
an	O
invertible	O
matrix	O
P	O
such	O
that	O
=	O
B	O
.	O
</s>
<s>
Compare	O
with	O
similar	O
matrices.EP	O
matrix	O
or	O
Range-Hermitian	O
matrixA	O
square	O
matrix	O
that	O
commutes	O
with	O
its	O
Moore	O
–	O
Penrose	O
inverse	O
:	O
AA+	O
=	O
A+A	O
.	O
</s>
<s>
Idempotent	O
matrix	O
or	O
Projection	B-Algorithm
Matrix	I-Algorithm
A	O
matrix	O
that	O
has	O
the	O
property	O
A²	O
=	O
AA	O
=	O
A	O
.	O
</s>
<s>
The	O
name	O
projection	B-Algorithm
matrix	I-Algorithm
inspires	O
from	O
the	O
observation	O
of	O
projection	B-Algorithm
of	O
a	O
point	O
multiple	O
times	O
onto	O
a	O
subspace(plane or a line )	O
giving	O
the	O
same	O
result	O
as	O
one	O
projection	B-Algorithm
.	O
</s>
<s>
Invertible	O
matrix	O
A	O
square	O
matrix	O
having	O
a	O
multiplicative	O
inverse	O
,	O
that	O
is	O
,	O
a	O
matrix	O
B	O
such	O
that	O
AB	O
=	O
BA	O
=	O
I	O
.	O
Invertible	O
matrices	B-Architecture
form	O
the	O
general	O
linear	O
group	O
.	O
</s>
<s>
Involutory	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
which	O
is	O
its	O
own	O
inverse	O
,	O
i.e.	O
,	O
AA	O
=	O
I	O
.	O
</s>
<s>
Signature	B-Algorithm
matrices	I-Algorithm
,	O
Householder	O
matrices	B-Architecture
(	O
Also	O
known	O
as	O
'	O
reflection	B-Algorithm
matrices	B-Architecture
 '	O
to	O
reflect	O
a	O
point	O
about	O
a	O
plane	O
or	O
line	O
)	O
have	O
this	O
property	O
.	O
</s>
<s>
Isometric	O
matrix	O
A	O
matrix	O
that	O
preserves	O
distances	O
,	O
i.e.	O
,	O
a	O
matrix	O
that	O
satisfies	O
A*A	O
=	O
I	O
where	O
A*	O
denotes	O
the	O
conjugate	B-Algorithm
transpose	I-Algorithm
of	O
A	O
.	O
Nilpotent	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
satisfying	O
Aq	O
=	O
0	O
for	O
some	O
positive	B-Algorithm
integer	O
q	O
.	O
Equivalently	O
,	O
the	O
only	O
eigenvalue	O
of	O
A	B-Application
is	I-Application
0	O
.	O
</s>
<s>
Normal	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
that	O
commutes	O
with	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
:	O
AA∗	O
=	O
A∗A	O
They	O
are	O
the	O
matrices	B-Architecture
to	O
which	O
the	O
spectral	O
theorem	O
applies	O
.	O
</s>
<s>
Orthogonal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
inverse	O
is	O
equal	O
to	O
its	O
transpose	O
,	O
A−1	O
=	O
AT	O
.	O
</s>
<s>
Orthonormal	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
columns	O
are	O
orthonormal	B-Algorithm
vectors	I-Algorithm
.	O
</s>
<s>
Partially	O
Isometric	O
matrix	O
A	O
matrix	O
that	O
is	O
an	O
isometry	O
on	O
the	O
orthogonal	B-Algorithm
complement	I-Algorithm
of	O
its	O
kernel	B-Algorithm
.	O
</s>
<s>
Unimodular	B-Algorithm
matrix	I-Algorithm
An	O
invertible	O
matrix	O
with	O
entries	O
in	O
the	O
integers	O
(	O
integer	B-Algorithm
matrix	I-Algorithm
)	O
Necessarily	O
the	O
determinant	O
is	O
+1	O
or	O
−1	O
.	O
</s>
<s>
See	O
also	O
unipotent	O
group.Unitary	O
matrixA	O
square	O
matrix	O
whose	O
inverse	O
is	O
equal	O
to	O
its	O
conjugate	B-Algorithm
transpose	I-Algorithm
,	O
A−1	O
=	O
A*	O
.	O
</s>
<s>
Totally	B-Algorithm
unimodular	I-Algorithm
matrix	O
A	O
matrix	O
for	O
which	O
every	O
non-singular	O
square	O
submatrix	O
is	O
unimodular	B-Algorithm
.	O
</s>
<s>
This	O
has	O
some	O
implications	O
in	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
of	O
an	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Weighing	O
matrix	O
A	O
square	O
matrix	O
the	O
entries	O
of	O
which	O
are	O
in	O
,	O
such	O
that	O
AAT	O
=	O
wI	O
for	O
some	O
positive	B-Algorithm
integer	O
w	O
.	O
</s>
<s>
Name	O
Explanation	O
Notes	O
Convergent	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
whose	O
successive	O
powers	O
approach	O
the	O
zero	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Defective	O
matrix	O
A	O
square	O
matrix	O
that	O
does	O
not	O
have	O
a	O
complete	O
basis	O
of	O
eigenvectors	O
,	O
and	O
is	O
thus	O
not	O
diagonalizable	B-Algorithm
.	O
</s>
<s>
Diagonalizable	B-Algorithm
matrix	I-Algorithm
A	O
square	O
matrix	O
similar	B-Algorithm
to	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Hurwitz	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
whose	O
eigenvalues	O
have	O
strictly	O
negative	O
real	O
part	O
.	O
</s>
<s>
A	O
stable	O
system	O
of	O
differential	O
equations	O
may	O
be	O
represented	O
by	O
a	O
Hurwitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
M-matrix	B-Algorithm
A	O
Z-matrix	B-Algorithm
with	O
eigenvalues	O
whose	O
real	O
parts	O
are	O
nonnegative	O
.	O
</s>
<s>
Positive-definite	B-Algorithm
matrix	I-Algorithm
A	O
Hermitian	B-Algorithm
matrix	I-Algorithm
with	O
every	O
eigenvalue	O
positive	B-Algorithm
.	O
</s>
<s>
Stability	B-Algorithm
matrix	I-Algorithm
Synonym	O
for	O
Hurwitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Stieltjes	O
matrix	O
A	O
real	O
symmetric	B-Algorithm
positive	I-Algorithm
definite	I-Algorithm
matrix	O
with	O
nonpositive	O
off-diagonal	O
entries	O
.	O
</s>
<s>
Special	O
case	O
of	O
an	O
M-matrix	B-Algorithm
.	O
</s>
<s>
Name	O
Definition	O
Comments	O
Adjugate	O
matrix	O
Transpose	O
of	O
the	O
cofactor	O
matrix	O
The	O
inverse	O
of	O
a	O
matrix	O
is	O
its	O
adjugate	O
matrix	O
divided	O
by	O
its	O
determinant	O
Augmented	B-Algorithm
matrix	I-Algorithm
Matrix	O
whose	O
rows	O
are	O
concatenations	O
of	O
the	O
rows	O
of	O
two	O
smaller	O
matrices	B-Architecture
Used	O
for	O
performing	O
the	O
same	O
row	O
operations	O
on	O
two	O
matrices	B-Architecture
Bézout	B-Algorithm
matrix	I-Algorithm
Square	O
matrix	O
whose	O
determinant	O
is	O
the	O
resultant	O
of	O
two	O
polynomials	O
See	O
also	O
Sylvester	B-Algorithm
matrix	I-Algorithm
Carleman	B-Algorithm
matrix	I-Algorithm
Infinite	O
matrix	O
of	O
the	O
Taylor	O
coefficients	O
of	O
an	O
analytic	B-Language
function	I-Language
and	O
its	O
integer	O
powers	O
The	O
composition	O
of	O
two	O
functions	O
can	O
be	O
expressed	O
as	O
the	O
product	O
of	O
their	O
Carleman	O
matrices	B-Architecture
Cartan	O
matrix	O
A	O
matrix	O
associated	O
with	O
either	O
a	O
finite-dimensional	O
associative	O
algebra	O
,	O
or	O
a	O
semisimple	O
Lie	O
algebra	O
Cofactor	O
matrix	O
Formed	O
by	O
the	O
cofactors	O
of	O
a	O
square	O
matrix	O
,	O
that	O
is	O
,	O
the	O
signed	O
minors	O
,	O
of	O
the	O
matrix	O
Transpose	O
of	O
the	O
Adjugate	O
matrix	O
Companion	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
having	O
the	O
coefficients	O
of	O
a	O
polynomial	O
as	O
last	O
column	O
,	O
and	O
having	O
the	O
polynomial	O
as	O
its	O
characteristic	O
polynomial	O
Coxeter	O
matrix	O
A	O
matrix	O
which	O
describes	O
the	O
relations	O
between	O
the	O
involutions	B-Algorithm
that	O
generate	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
Distance	O
matrix	O
The	O
square	O
matrix	O
formed	O
by	O
the	O
pairwise	O
distances	O
of	O
a	O
set	O
of	O
points	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
is	O
a	O
special	O
case	O
Euclidean	B-Algorithm
distance	I-Algorithm
matrix	I-Algorithm
A	O
matrix	O
that	O
describes	O
the	O
pairwise	O
distances	O
between	O
points	O
in	O
Euclidean	O
space	O
See	O
also	O
distance	O
matrix	O
Fundamental	B-General_Concept
matrix	I-General_Concept
The	O
matrix	O
formed	O
from	O
the	O
fundamental	O
solutions	O
of	O
a	O
system	O
of	O
linear	O
differential	O
equations	O
Generator	O
matrix	O
In	O
Coding	B-Error_Name
theory	I-Error_Name
,	O
a	O
matrix	O
whose	O
rows	O
span	O
a	O
linear	O
code	O
Gramian	B-Algorithm
matrix	I-Algorithm
The	O
symmetric	B-Algorithm
matrix	I-Algorithm
of	O
the	O
pairwise	O
inner	O
products	O
of	O
a	O
set	O
of	O
vectors	O
in	O
an	O
inner	O
product	O
space	O
Hessian	O
matrix	O
The	O
square	O
matrix	O
of	O
second	O
partial	O
derivatives	O
of	O
a	O
function	O
of	O
several	O
variables	O
Householder	O
matrix	O
The	O
matrix	O
of	O
a	O
reflection	B-Algorithm
with	O
respect	O
to	O
a	O
hyperplane	O
passing	O
through	O
the	O
origin	O
Jacobian	O
matrix	O
The	O
matrix	O
of	O
the	O
partial	O
derivatives	O
of	O
a	O
function	O
of	O
several	O
variables	O
Moment	B-Algorithm
matrix	I-Algorithm
Used	O
in	O
statistics	O
and	O
Sum-of-squares	O
optimization	O
Payoff	O
matrix	O
A	O
matrix	O
in	O
game	O
theory	O
and	O
economics	O
,	O
that	O
represents	O
the	O
payoffs	O
in	O
a	O
normal	O
form	O
game	O
where	O
players	O
move	O
simultaneously	O
Pick	O
matrix	O
A	O
matrix	O
that	O
occurs	O
in	O
the	O
study	O
of	O
analytical	O
interpolation	O
problems	O
Rotation	B-Algorithm
matrix	I-Algorithm
A	O
matrix	O
representing	O
a	O
rotation	B-General_Concept
Seifert	O
matrix	O
A	O
matrix	O
in	O
knot	O
theory	O
,	O
primarily	O
for	O
the	O
algebraic	O
analysis	O
of	O
topological	O
properties	O
of	O
knots	O
and	O
links	O
.	O
</s>
<s>
The	O
following	O
matrices	B-Architecture
find	O
their	O
main	O
application	O
in	O
statistics	O
and	O
probability	O
theory	O
.	O
</s>
<s>
Centering	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
which	O
,	O
when	O
multiplied	O
with	O
a	O
vector	O
,	O
has	O
the	O
same	O
effect	O
as	O
subtracting	O
the	O
mean	O
of	O
the	O
components	O
of	O
the	O
vector	O
from	O
every	O
component	O
.	O
</s>
<s>
Correlation	O
matrix	O
—	O
a	O
symmetric	B-Algorithm
n×n	O
matrix	O
,	O
formed	O
by	O
the	O
pairwise	O
correlation	O
coefficients	O
of	O
several	O
random	O
variables	O
.	O
</s>
<s>
Covariance	O
matrix	O
—	O
a	O
symmetric	B-Algorithm
n×n	O
matrix	O
,	O
formed	O
by	O
the	O
pairwise	O
covariances	O
of	O
several	O
random	O
variables	O
.	O
</s>
<s>
Hat	B-Algorithm
matrix	I-Algorithm
—	O
a	O
square	O
matrix	O
used	O
in	O
statistics	O
to	O
relate	O
fitted	O
values	O
to	O
observed	O
values	O
.	O
</s>
<s>
Precision	B-General_Concept
matrix	I-General_Concept
—	O
a	O
symmetric	B-Algorithm
n×n	O
matrix	O
,	O
formed	O
by	O
inverting	O
the	O
covariance	O
matrix	O
.	O
</s>
<s>
Stochastic	B-Algorithm
matrix	I-Algorithm
—	O
a	O
non-negative	B-Algorithm
matrix	I-Algorithm
describing	O
a	O
stochastic	O
process	O
.	O
</s>
<s>
The	O
following	O
matrices	B-Architecture
find	O
their	O
main	O
application	O
in	O
graph	O
and	O
network	O
theory	O
.	O
</s>
<s>
Adjacency	B-Algorithm
matrix	I-Algorithm
—	O
a	O
square	O
matrix	O
representing	O
a	O
graph	O
,	O
with	O
aij	O
non-zero	O
if	O
vertex	O
i	O
and	O
vertex	O
j	O
are	O
adjacent	O
.	O
</s>
<s>
Biadjacency	O
matrix	O
—	O
a	O
special	O
class	O
of	O
adjacency	B-Algorithm
matrix	I-Algorithm
that	O
describes	O
adjacency	O
in	O
bipartite	O
graphs	O
.	O
</s>
<s>
Degree	B-Algorithm
matrix	I-Algorithm
—	O
a	O
diagonal	B-Algorithm
matrix	I-Algorithm
defining	O
the	O
degree	O
of	O
each	O
vertex	O
in	O
a	O
graph	O
.	O
</s>
<s>
Incidence	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
representing	O
a	O
relationship	O
between	O
two	O
classes	O
of	O
objects	O
(	O
usually	O
vertices	O
and	O
edges	O
in	O
the	O
context	O
of	O
graph	O
theory	O
)	O
.	O
</s>
<s>
Laplacian	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
equal	O
to	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
minus	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
for	O
a	O
graph	O
,	O
used	O
to	O
find	O
the	O
number	O
of	O
spanning	O
trees	O
in	O
the	O
graph	O
.	O
</s>
<s>
Seidel	O
adjacency	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
similar	B-Algorithm
to	O
the	O
usual	O
adjacency	B-Algorithm
matrix	I-Algorithm
but	O
with	O
1	O
for	O
adjacency	O
;	O
+1	O
for	O
nonadjacency	O
;	O
0	O
on	O
the	O
diagonal	O
.	O
</s>
<s>
Skew-adjacency	O
matrix	O
—	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
in	O
which	O
each	O
non-zero	O
aij	O
is	O
1	O
or	O
1	O
,	O
accordingly	O
as	O
the	O
direction	O
i	O
→	O
j	O
matches	O
or	O
opposes	O
that	O
of	O
an	O
initially	O
specified	O
orientation	O
.	O
</s>
<s>
Cabibbo	O
–	O
Kobayashi	O
–	O
Maskawa	O
matrix	O
—	O
a	O
unitary	B-Algorithm
matrix	I-Algorithm
used	O
in	O
particle	O
physics	O
to	O
describe	O
the	O
strength	O
of	O
flavour-changing	O
weak	O
decays	O
.	O
</s>
<s>
Hermitian	B-Algorithm
,	O
non-negative	B-Algorithm
and	O
with	O
trace	O
1	O
.	O
</s>
<s>
Fundamental	B-General_Concept
matrix	I-General_Concept
(	O
computer	B-Application
vision	I-Application
)	O
—	O
a	O
3	O
×	O
3	O
matrix	O
in	O
computer	B-Application
vision	I-Application
that	O
relates	O
corresponding	O
points	O
in	O
stereo	O
images	O
.	O
</s>
<s>
Fuzzy	B-Algorithm
associative	I-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
in	O
artificial	B-Application
intelligence	I-Application
,	O
used	O
in	O
machine	O
learning	O
processes	O
.	O
</s>
<s>
Gamma	O
matrices	B-Architecture
—	O
4	O
×	O
4	O
matrices	B-Architecture
in	O
quantum	O
field	O
theory	O
.	O
</s>
<s>
Gell-Mann	O
matrices	B-Architecture
—	O
a	O
generalization	O
of	O
the	O
Pauli	O
matrices	B-Architecture
;	O
these	O
matrices	B-Architecture
are	O
one	O
notable	O
representation	O
of	O
the	O
infinitesimal	O
generators	O
of	O
the	O
special	O
unitary	O
group	O
SU(3 )	O
.	O
</s>
<s>
Hamiltonian	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
used	O
in	O
a	O
variety	O
of	O
fields	O
,	O
including	O
quantum	O
mechanics	O
and	O
linear-quadratic	O
regulator	O
(	O
LQR	O
)	O
systems	O
.	O
</s>
<s>
Irregular	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
used	O
in	O
computer	B-General_Concept
science	I-General_Concept
which	O
has	O
a	O
varying	O
number	O
of	O
elements	O
in	O
each	O
row	O
.	O
</s>
<s>
Overlap	O
matrix	O
—	O
a	O
type	O
of	O
Gramian	B-Algorithm
matrix	I-Algorithm
,	O
used	O
in	O
quantum	O
chemistry	O
to	O
describe	O
the	O
inter-relationship	O
of	O
a	O
set	O
of	O
basis	O
vectors	O
of	O
a	O
quantum	O
system	O
.	O
</s>
<s>
State	O
transition	B-Algorithm
matrix	I-Algorithm
—	O
exponent	O
of	O
state	O
matrix	O
in	O
control	O
systems	O
.	O
</s>
<s>
Substitution	B-Algorithm
matrix	I-Algorithm
—	O
a	O
matrix	O
from	O
bioinformatics	O
,	O
which	O
describes	O
mutation	O
rates	O
of	O
amino	O
acid	O
or	O
DNA	O
sequences	O
.	O
</s>
<s>
Supnick	B-Algorithm
matrix	I-Algorithm
—	O
a	O
square	O
matrix	O
used	O
in	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
Z-matrix	B-Algorithm
—	O
a	O
matrix	O
in	O
chemistry	O
,	O
representing	O
a	O
molecule	O
in	O
terms	O
of	O
its	O
relative	O
atomic	O
geometry	O
.	O
</s>
<s>
Wilson	B-Algorithm
matrix	I-Algorithm
,	O
a	O
matrix	O
used	O
as	O
an	O
example	O
for	O
test	O
purposes	O
.	O
</s>
<s>
Pseudoinverse	B-Algorithm
—	O
a	O
generalization	O
of	O
the	O
inverse	O
matrix	O
.	O
</s>
<s>
Row	O
echelon	O
form	O
—	O
a	O
matrix	O
in	O
this	O
form	O
is	O
the	O
result	O
of	O
applying	O
the	O
forward	O
elimination	O
procedure	O
to	O
a	O
matrix	O
(	O
as	O
used	O
in	O
Gaussian	B-Algorithm
elimination	I-Algorithm
)	O
.	O
</s>
