A matrix -algebra is a nonempty olle tion A C nn
losed under sums, s alar and
matrix multipli ation, and takingthe adjoint. Call A;A 0
C nn
equivalentifthere exists
a unitary matrixU 2C nn
su hthat
(1) A
0
=fU
MU jM 2Ag:
Formatri esM
1
andM
2
,the dire tsum is
(2) M
1
M
2 :=
M
1 0
0 M
2
:
The iterateddire tsum of M
1
;:::;M
n
isdenoted by
(3)
n
M
i=1 M
i :
Wewrite
(4) tM :=
t
M
i=1 M:
Call Abasi if
(5) A=tC mm
:=ftM jM 2C mm
g
forsomet andm.
Thedire tsum ofA and A 0
is
(6) AA
0
:=fMM 0
jM 2A;M 0
2A 0
g:
A is alled azero algebra ifAonly onsistsof thezero matrix.
Theorem. Ea h matrix-algebrais equivalent toa dire tsum of basi algebrasand a zero
algebra.
Proof. We rst show:
(7) Ea h matrix -algebra A is equivalent to a dire t sum of a matrix -algebra
ontaining the identity matrix and a zero algebra. In parti ular, A ontains a
unit.
Let N be a matrix inA of maximum rank. Then the row spa e rowN of N ontains the
row spa eof ea h matrixinA. Forlet M 2A. ThenkerM kerN,sin e
(8) x2ker(M M+N N) () (M M +N N)x=0 () x (M M +N N)x=
0 () x
M
Mx=0 and x
N
Nx=0 () Mx=0 and Nx=0 ()
x2kerM\kerN.
By themaximalityof therank ofN,ker(M
M +N
N)=kerN,hen e ker(M)kerN.
So the row spa e of ea h matrix in A is ontained in rowN = row(N
N). We an
assumethatN
N isa diagonalmatrix. (Repla eAbyfU
MU jM 2Agforsomeunitary
matrix U.) So A is a dire t sum of a zero algebra and a matrix -algebra ontaining a
nonsingular diagonal matrix . Then I is a linear ombination of ; 2
; 3
;::: (by the
theory ofVandermonde matri es).
Thisproves (7). Hen e toprove thetheorem, we anassumethat I 2A.
LetC
A
bethe enterof A;thatis,
(9) C
A
:=fC2AjCM =MC forallM 2Ag.
As C
A
is a ommutative matrix -algebra, the matri es in C
A
an be simultaneously di-
agonalized by some unitary matrix U. That is, fU
MU j M 2 C
A
g onsists of diagonal
matri es. So we an assume that C
A
onsists of diagonal matri es only. Then C
A is the
linear hull of ertaindiagonal 0,1matri es E
1
;:::;E
t
,withE
i E
j
=0ifi6=j.
Thenforea hM 2Aandea hionehasE
i
M =ME
i
. SoM is0inpositions(k;l)with
(E
i )
k;k 6=(E
i )
l:l
=0. So A is the dire t sum of matrix-algebras ea h with the property
that the s alar multiples of the identity matrix (of appropriate dimension) are the only
matri es ommuting withall matri esin thesubalgebra.
Hen eit suÆ estoshow
(10) ifAis amatrix -algebrawithC
A
=CI,then Ais basi .
Let B be a maximal subset of A with the property that A
= A and AB = BA for all
A;B 2B. ThisimpliesthatB isa ommutative-subalgebra ofAandthatI 2B. We an
assumethatB onsistsofdiagonalmatri esonly. LetE
1
;:::;E
t
betheminimalidempotents
of B. So B=CE
1
++CE
t andE
1
++E
t
=I.
Nowforea h i:
(11) Forea h M 2Athere is a 2C withE
i ME
i
=E
i
. Inother words, E
i AE
i
=
CE
i .
Suppose not. We an assume that M
= M, sin e we an repla e M by M +M
or
iM iM
. Thenwe anadd E
i ME
i
toB,sin eit ommuteswithea h ofE
1
;:::;E
t . This
ontradi tsthe maximalityofB. This proves(11).
Moreover,
(12) forall ithere isan M
i
2AwithE
1 M
i E
i 6=0.
Tosee this, onsider thelinear spa e I generated byAE
1
A. Thisis a-algebra. LetP be
PM =PMP =MP. SoP 2C
A
. Hen e bytheassumption in(10),P =I. So I =A,and
thereforeE
i
2I,and hen eE
i
isasumofmatri esinAE
1
A,hen einAE
1 AE
i
. Hen e we
have (12).
We anassume that M
i
=E
1 M
i E
i
(byresetting M
i := E
1 M
i E
i
). NowM
i M
i
belongs
toE
i AE
i
,hen e by(11) toCE
i
. Similarly, M
i M
i 2CE
1
. By s aling we anassume that
M
i M
i
=E
i
forea h i. This implies that M
i M
i
= E
1
for ea h i, sin e M
i M
i
= E
1 for
some2C. Then
(13)
2
E
1
=M
i M
i M
i M
i
=M
i E
i M
i
=M
i M
i
=E
1
;
and hen e =1. Sin e M
1 2E
1 AE
1
,wehave by(11) thatM
1
=E
1
forsome2C. As
M
1 M
1
=E
1
,we know=1. Hen erepla ingM
1 by
1
E
1
,weobtain M
1
=E
1 .
So rank(E
1
) = rank(M
i M
i
) = rank(M
i M
i
) =rank(E
i
) for ea h i. Hen e all E
i have
the same number of 1's. ForA 2 A and i;j =1;:::;t,let A
i;j
be the submatrix indu ed
bytherowswhereE
i
has1'sand the olumnswhereE
j
has1's. So E
i AE
j
arisesfrom A
i;j
byadding all-zerorows and olumns.
So for ea h i, the matrix (M
i )
1;i
is unitary. Let U
i
:= (M
i )
1;i
. Let U be the unitary
matrix with U
i;i
= U
i
forea h i, and U
i;j
=0 ifi 6=j. Repla ing A byU
AU,we obtain
that (M
i )
1;i
=I forea h i.
Nowfori;j=1;:::;t,let N
i;j
be be thematrixwithE
i N
i;j E
j
=N
i;j
and (N
i;j )
i;j
=I.
So N
i;i
= E
i
and N
1;i
= M
i
for ea h i = 1;:::;t. So ea h N
1;i
belongs to A. Sin e
N
i;j
=N
1;i N
1;j
forall i;j,it follows that N
i;j
2A forall i;j.
We nallyshow:
(14) forea h M 2A and i;j=1;:::;t,E
i ME
j
=N
i;j
forsome2C.
Indeed, using (11),
(15) E
i ME
j
=E
i ME
j N
j;i E
i N
i;j 2E
i AE
i N
i;j
=CE
i N
i;j
=CN
i;j
;
asrequired. So Ais basi .