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Algebra of complex square matrices
A coherent algebra is an algebra of complex square matrices that is closed under ordinary matrix multiplication , Schur product , transposition , and contains both the identity matrix
I
{\displaystyle I}
J
{\displaystyle J}
[ 1]
A subspace
A
{\displaystyle {\mathcal {A}}}
M
a
t
n
×
n
(
C
)
{\displaystyle \mathrm {Mat} _{n\times n}(\mathbb {C} )}
n
{\displaystyle n}
I
,
J
∈
A
{\displaystyle I,J\in {\mathcal {A}}}
M
T
∈
A
{\displaystyle M^{T}\in {\mathcal {A}}}
M
∈
A
{\displaystyle M\in {\mathcal {A}}}
M
N
∈
A
{\displaystyle MN\in {\mathcal {A}}}
M
∘
N
∈
A
{\displaystyle M\circ N\in {\mathcal {A}}}
M
,
N
∈
A
{\displaystyle M,N\in {\mathcal {A}}}
A coherent algebra
A
{\displaystyle {\mathcal {A}}}
Homogeneous if every matrix in
A
{\displaystyle {\mathcal {A}}}
Commutative if
A
{\displaystyle {\mathcal {A}}}
Symmetric if every matrix in
A
{\displaystyle {\mathcal {A}}}
The set
Γ
(
A
)
{\displaystyle \Gamma ({\mathcal {A}})}
Schur-primitive matrices in a coherent algebra
A
{\displaystyle {\mathcal {A}}}
Γ
(
A
)
:=
{
M
∈
A
:
M
∘
M
=
M
,
M
∘
N
∈
span
{
M
}
for all
N
∈
A
}
{\displaystyle \Gamma ({\mathcal {A}}):=\{M\in {\mathcal {A}}:M\circ M=M,M\circ N\in \operatorname {span} \{M\}{\text{ for all }}N\in {\mathcal {A}}\}}
Dually, the set
Λ
(
A
)
{\displaystyle \Lambda ({\mathcal {A}})}
primitive matrices in a coherent algebra
A
{\displaystyle {\mathcal {A}}}
Λ
(
A
)
:=
{
M
∈
A
:
M
2
=
M
,
M
N
∈
span
{
M
}
for all
N
∈
A
}
{\displaystyle \Lambda ({\mathcal {A}}):=\{M\in {\mathcal {A}}:M^{2}=M,MN\in \operatorname {span} \{M\}{\text{ for all }}N\in {\mathcal {A}}\}}
The centralizer of a group of permutation matrices is a coherent algebra, i.e.
W
{\displaystyle {\mathcal {W}}}
n
{\displaystyle n}
W
:=
{
M
∈
M
a
t
n
×
n
(
C
)
:
M
P
=
P
M
for all
P
∈
S
}
{\displaystyle {\mathcal {W}}:=\{M\in \mathrm {Mat} _{n\times n}(\mathbb {C} ):MP=PM{\text{ for all }}P\in S\}}
S
{\displaystyle S}
n
×
n
{\displaystyle n\times n}
group of permutation matrices representing the automorphism group of a graph
G
{\displaystyle G}
G
{\displaystyle G}
vertex-transitive .[ 2]
The span of the set of matrices relating pairs of elements lying in the same orbit of a diagonal action of a finite group on a finite set is a coherent algebra, i.e.
W
:=
span
{
A
(
u
,
v
)
:
u
,
v
∈
V
}
{\displaystyle {\mathcal {W}}:=\operatorname {span} \{A(u,v):u,v\in V\}}
A
(
u
,
v
)
∈
Mat
V
×
V
(
C
)
{\displaystyle A(u,v)\in \operatorname {Mat} _{V\times V}(\mathbb {C} )}
(
A
(
u
,
v
)
)
x
,
y
:=
{
1
if
(
x
,
y
)
=
(
u
g
,
v
g
)
for some
g
∈
G
0
otherwise
{\displaystyle (A(u,v))_{x,y}:={\begin{cases}1\ {\text{if }}(x,y)=(u^{g},v^{g}){\text{ for some }}g\in G\\0{\text{ otherwise }}\end{cases}}}
u
,
v
∈
V
{\displaystyle u,v\in V}
V
{\displaystyle V}
G
{\displaystyle G}
The span of a regular representation of a finite group as a group of permutation matrices over
C
{\displaystyle \mathbb {C} }
The intersection of a set of coherent algebras of order
n
{\displaystyle n}
The tensor product of coherent algebras is a coherent algebra, i.e.
A
⊗
B
:=
{
M
⊗
N
:
M
∈
A
and
N
∈
B
}
{\displaystyle {\mathcal {A}}\otimes {\mathcal {B}}:=\{M\otimes N:M\in {\mathcal {A}}{\text{ and }}N\in {\mathcal {B}}\}}
A
∈
Mat
m
×
m
(
C
)
{\displaystyle {\mathcal {A}}\in \operatorname {Mat} _{m\times m}(\mathbb {C} )}
B
∈
M
a
t
n
×
n
(
C
)
{\displaystyle {\mathcal {B}}\in \mathrm {Mat} _{n\times n}(\mathbb {C} )}
The symmetrization
A
^
:=
span
{
M
+
M
T
:
M
∈
A
}
{\displaystyle {\widehat {\mathcal {A}}}:=\operatorname {span} \{M+M^{T}:M\in {\mathcal {A}}\}}
A
{\displaystyle {\mathcal {A}}}
If
A
{\displaystyle {\mathcal {A}}}
M
T
∈
Γ
(
A
)
{\displaystyle M^{T}\in \Gamma ({\mathcal {A}})}
M
∈
A
{\displaystyle M\in {\mathcal {A}}}
A
=
span
(
Γ
(
A
)
)
{\displaystyle {\mathcal {A}}=\operatorname {span} \left(\Gamma ({\mathcal {A}}\right))}
I
∈
Γ
(
A
)
{\displaystyle I\in \Gamma ({\mathcal {A}})}
A
{\displaystyle {\mathcal {A}}}
Dually, if
A
{\displaystyle {\mathcal {A}}}
n
{\displaystyle n}
E
T
,
E
∗
∈
Λ
(
A
)
{\displaystyle E^{T},E^{*}\in \Lambda ({\mathcal {A}})}
E
∈
A
{\displaystyle E\in {\mathcal {A}}}
1
n
J
∈
Λ
(
A
)
{\displaystyle {\frac {1}{n}}J\in \Lambda ({\mathcal {A}})}
A
=
span
(
Λ
(
A
)
)
{\displaystyle {\mathcal {A}}=\operatorname {span} \left(\Lambda ({\mathcal {A}}\right))}
Every symmetric coherent algebra is commutative, and every commutative coherent algebra is homogeneous.
A coherent algebra is commutative if and only if it is the Bose–Mesner algebra of a (commutative) association scheme .[ 1]
A coherent algebra forms a principal ideal ring under Schur product; moreover, a commutative coherent algebra forms a principal ideal ring under ordinary matrix multiplication as well. 
