Abstract -This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator
[
A,B]
=0 for two matrices A and B if and only if a vector v (B) defined uniquely from the matrix B is in the null space of a well- structured matrix defined as the Kronecker sum A⊕( )
−A* , which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning commutators of certain matrices with functions of matrices f( )
A which extends the well- known sufficiency – type commuting result[
A,f( )
A]
=0.Index Terms- Commuting matrices, Jordan canonical form, Kronecker product, Kronecker sum, Function of a matrix. AMS Classification Subjects: 15A30, 11C20
I. INTRODUCTION
The problem of commuting matrices is very relevant in certain problems of Engineering and Physics . In particular, such a problem is of crucial interest related to discrete Fourier transforms, normal modes in dynamic systems or commuting matrices dependent on a parameter (see, for instance, [1-3]). It is well-known that commuting matrices have at least a common eigenvector and also, a common generalized eigenspace, [4-5]. A less restrictive problem of interest in the above context is that of almost commuting matrices, roughly speaking, the norm of the commutator is sufficiently small, [5-6]. A very relevant related result is that the sum of matrices which commute is an infinitesimal generator of a
0
C - semigroup. This leads to a well-known result in Systems Theory establishing that that the matrix function
2 2 1 1 2 2 1
1t A t A t A t
A
e e
e + = is a fundamental (or state
transition) matrix for the cascade of the time invariant differential systems x&1
( )
t =A1x1( )
t , operating on a time1
t , and x&2
( )
t =A2 x2( )
t , operating on a time t2 , provided that A1 and A2 commute (see, [7-11] ).The problem of commuting matrices is also of relevant interest in dynamic switched systems, namely, those which possess several parameterizations one of each is activated at each current time interval. If the matrices of dynamics of all the parameterizations commute then there exists a common Lyapunov function for all those parameterizations and anyManuscript received October 11, 2008. This work was supported in part by the Spanish Ministry of Education through project DPI2006-00714. It was also supported in part the Basque Government through Grants GIC07143-IT-269-07, SAIOTEK S-PE08UN15.
The author is with the Institute of Research and Development of Processes Campus of Leio (Bizkaia). Aptdo. 644- Bilbao. Spain (e-mail: [email protected]).
arbitrary switching rule operating at any time instant maintains the global stability of the switched rule provided that all the parameterizations are stable, [7]. However, in the case that there is no common Lyapunov function for all the parameterizations, a minimum residence time at each active parameterization is needed to maintain the global stability of the switched system so that the switching rule among distinct parameterizations is not arbitrary, [12-13]. Parallel results apply for switched time-delay systems subject to point delays under zero or sufficiently small delays when the matrices defining the delay-free dynamics of the various parameterizations commute, [10-11]. This paper formulates the necessary and sufficient condition for any countable set of (real or complex) matrices to commute.
1.1. Notation
[
A,B]
is the commutator of the square matrices A and B.(
a B)
:B
A⊗ = ij is the Kronecker (or direct) product of
( )
aij :A = and B.
B I I A : B
A⊕ = ⊗ n+ n⊗ is the Kronecker sum of the square matrices A:=
( )
aij and both of order n, where Inis the n-th identity matrix.T
A is the transpose of the matrix A and A*is the conjugate transpose of the complex matrix A. For any matrix A, ImAand Ker A are its associate range (or image) subspace and null space, respectively. Also, rank (A) is the rank of A which is the dimension of Im (A) and det (A) is the determinant of the square matrix A.
( )
(
)
2n T T n T 2 T
1,a ,....,a a
A
v = ∈C if aiT:=
(
ai1,ai2,....,ain)
is the i-th row of the square matrix A.( )
Aσ is the spectrum of A ; n:=
{
1,2,...,n}
. If λi∈σ( )
Athen there exist positive integers μiand νi≤μi which are, respectively, its algebraic and geometric multiplicity; i.e. the times it is repeated in the characteristic polynomial of A and the number of its associate Jordan blocks, respectively. The integer μ≤nis the number of distinct eigenvalues and the integer mi, subject to
i i m
1≤ ≤μ , is the index of λi∈σ
( )
A ; ∀i∈μ, that is, its multiplicity in the minimal polynomial of A.A ∼ B denotes a similarity transformation from A to B=T−1AT for given A,B∈R n×n for some non-singular T∈R n×n. A
≈ B = E A F means that there is an equivalence transformation for
given A,B∈R n×n for some non-singular E,F∈Rn×n. A linear transformation from Rnto Rn, represented by the matrix T∈R n×n, is denoted identically to such a matrix in order to simplify the notation. If V≠DomT≡Rn is a subspace of
n
R then ImT
( ) {
V:= Tz:z∈V}
andThe Necessary and Sufficient Condition for a Set
of Matrices to Commute and Related Results
( )
V:{
z V : Tz 0 n}
TKer = ∈ = ∈R . IfV≡Rn, the notation
is simplified to ImT:=
{
Tz: z∈Rn}
and{
z n :Tz 0 n}
: T
Ker = ∈R = ∈R .
The symbols “∧ ” and “∨” stand for logic conjunction and disjunction, respectively.
The abbreviation “iff ” stands for “ if and only if”. The notation card U stands for the cardinal of the set U.
II. COMMUTING AND NON – COMMUTING MATRICES WITH A GIVEN ONE
Consider the sets CA:=
{
X∈Rn×n:[
A,X]
=0}
≠∅, of matrices which commute with A, and[
]
{
: A,X 0}
:
C A = X∈Rn×n ≠ , of matrices which do not commute with A; ∀A∈Rn×n . Note that A
n n
C 0∈R × ∩ ; i.e. the zero n- matrix commutes with any n-matrix so that,
equivalently, A
n n
C
0∉R × ∩ and then CA∩CA=∅ ; n
n
A∈ ×
∀ R . The following two basic results follow concerning commutation and non- commutation of two matrices:
Propositions 2.1. (i)
( )
(
(
)
)
{
n n T}
A: :v X Ker A A
C = X∈R × ∈ ⊕ − , and
equivalently,
[
A,X]
=0⇔ v( )
X ∈Ker(
A⊕(
−AT)
)
. (ii)( )
(
(
)
)
{
n n T}
A n n
A: \C :v X Ker A A
C =R × = X∈R × ∉ ⊕ −
( )
(
(
)
)
{
n n T}
A A Im X v
: ∈ ⊕ −
∈
≡ X R × , and ,
equivalently,
[
A,X]
≠0⇔ v( )
X ∈Im(
A⊕(
−AT)
)
.(iii) B∈C A:=
{
X∈Rn×n:v( )
X ∈Ker(
A⊕( )
− AT)
}
( )
(
( )
)
{
n n T}
B: :v X Ker B B
C
A∈ = ∈ ∈ ⊕ −
⇔ X R ×
Proof: (i) –(ii) First note by inspection that
{
0,A}
CA⊃ ≠
∅ ; ∀A∈Rn×n . Also,
[
A,X]
=AX−XA=(
A⊗In −In ⊗AT)
v( )
X(
)
(
A⊕ −AT)
v( )
X =0 =( )
(
(
T)
)
A A Ker X
v ∈ ⊕ −
⇒ and Propositions 2.1(i)-(ii)
have been proved since there is an isomorphism n
n n2 :
f R ↔R × defined by f
(
v( )
X)
=X ; ∀X∈Rn×nfor
( )
(
T)
T n2n T 2 T
1,x ,....,x
x X
v = ∈R if
(
i1 i2 in)
T
i : x ,x ,....,x
x = is the i-th row of the square matrix X.
(iii) It is a direct consequence of Proposition 2.1 (iii) and the symmetry property of the commutator of two commuting matrices B∈CA ⇔
[
A,B] [
= B,A]
=0⇔A∈CB . Proposition 2.2.( )
(
A AT)
n2 Ker(
A( )
AT)
0 CArank ⊕ − < ⇔ ⊕ − ≠ ∈
(
)
(
A A)
X( )
00∈σ ⊕ − T ⇔∃ ≠
⇔ ; ∀A∈Rn×n .
Proof:
[
A,A]
=0;∀A∈Rn×n⇒( )
(
(
T)
)
n
A A Ker A v 0
2
− ⊕ ∈
≠ ∋
∃R ; ∀A∈Rn×n . As
a result,
(
)
(
T)
n20 A A
Ker ⊕ − ≠ ∈R ; ∀A∈Rn×n ⇔
(
)
(
T)
2n A A
rank ⊕ − < ; ∀A∈Rn×n
so that 0∈σ
(
A⊕( )
− AT)
.Also, ∃X
( )
≠0 ∈Rn×n :[
A,X]
=0 ⇔X∈CA since(
)
(
T)
n20 A A
Ker ⊕ − ≠ ∈R .
Then, Proposition 2.2 has been proved. The subsequent result is stronger than Proposition 2.2.
Theorem 2.3. The following properties hold:
(i) The spectrum of A⊕
(
− AT)
is(
)
(
A A)
{
ij i j: i, j( )
A ; i,j n}
T = λ =λ −λ λ λ ∈σ ∀ ∈
− ⊕ σ
and possesses ν Jordan blocks in its Jordan canonical form
of, subject to the constraints
( )
0 Sdim
n 2
1 i i
2 ≥ν
⎟ ⎟ ⎠ ⎞ ⎜
⎜ ⎝ ⎛
ν = = ν
≥ ∑μ
= , and
( )
(
T)
A A
0∈σ ⊕ − with an algebraic multiplicity
( )
0μ and with a geometric multiplicityν
( )
0 subject to the constraints:( )
0( )
0 nn
1 i
2 i 1
i 2 i 2
1
i i
2 ≥μ ≥ μ ≥ν = ν ≥
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
μ
= ∑μ
= ∑
μ = ∑μ
=
(2.1) where:
a) S:=span
{
zi ⊗x j,∀i,j∈n}
, μi=μ( )
λi and( )
i i=ν λν are, respectively, the algebraic and the geometric multiplicities of λi∈σ
( )
A , ∀i∈n; μ≤n is the number of distinct λi∈σ( )
A(
i∈μ)
, μi=μ( )
λij and( )
ij ji =ν λ
ν , are, respectively, the algebraic and the
geometric multiplicity of
(
)
(
(
T)
)
j i j
i = λ −λ ∈σ A⊕ −A
λ , ∀i,j∈n; μ≤n , and
b) xjand zi are, respectively, the right eigenvectors of A and AT with respective associated eigenvalues λj and λi ;∀i,j∈n .
(ii)
( )
(
A A)
rank(
A( )
A)
n( )
0 Imdim ⊕ − T = ⊕ − T = 2−ν
⇔ dimKer
(
A⊕( )
−AT)
= ν( )
0 ; nn
A∈ ×
∀ R (2.2) Proof: (i) Note that
( )
( )
TA A =σ
σ
⇒
(
)
(
A⊕ −AT)
:={
∋η=λk −λ ;∀λk,λ ∈σ( )
A ;∀k, ∈n}
σ C l l l
=σ0
(
A⊕( )
−AT)
∪σ0(
A⊕( )
−AT)
where( )
(
A AT)
{
(
A( )
AT)
: 0}
0 ⊕ − = λ∈σ ⊕ − λ=
(
)
(
A AT)
{
(
A(
AT)
)
: 0}
0 ⊕ − = λ∈σ ⊕ − λ≠
σ
(
)
(
)
(
(
T)
)
0
T \ A A
A
A⊕ − σ ⊕ −
σ =
Furthermore,
(
)
(
A⊕ −AT)
:={
∋λ=λ j−λi:λi,λj∈σ( )
A ;∀i,j∈n}
σ C
and zi⊗x j is a right eigenvector of A⊕
(
−AT)
associated with its eigenvalue λ ji =λ j−λi .
(
)
(
T)
i
j−λ ∈σ A⊕ −A
λ =
λ has algebraic and
geometric multiplicities μ ji and νji , respectively ; n
j , i ∈
∀ , since xj and zi are, respectively, the right eigenvectors of A and AT with associated eigenvalues λj and λi ;∀i,j∈n .
Let JA be the Jordan canonical form of A. It is first proved that there exists a non-singular
2 2
n n
T∈R × such that
(
)
(
A A)
TT J
J 1 T
A
A T⎟= ⊕ −
⎠ ⎞ ⎜
⎝ ⎛−
⊕ − . The proof is
made by direct verification by using the properties of the Kronecker product, with T=P ⊗PTfor a non-singular
n n
P∈R × such that A∼JA=P−1AP, as follows:
(
)
(
T) (
T)
11 A A T P P
T− ⊕ − = ⊗ −
(
)
(
) (
) (
T)(
T)
n 1 T T
n P P P P I A P P
I
A⊗ ⊗ − ⊗ ⊗ ⊗
× −
(
) (
) (
) (
T T T)
n 1 T
n T
1AP P I P P I P P A P
P− ⊗ − − − ⊗ −
=
=
(
P−1AP)
⊗In −In ⊗(
P−TATPT)
T A n nA I I J
J ⊗ − ⊗
=
⎟ ⎠ ⎞ ⎜
⎝ ⎛ − ⊗ + ⊗
= T
A n
n
A I I J
J ⎟
⎠ ⎞ ⎜
⎝ ⎛ − ⊕
= T
A
A J
J
and the result has been proved. Thus,
( )
(
)
⎟⎠ ⎞ ⎜
⎝
⎛ ⎟
⎠ ⎞ ⎜
⎝ ⎛ − ⊕ =
−
⊕ T
A A
T rank J J
A A
rank . It turns out
that P is, furthermore, unique except for multiplication by any nonzero real constant. Otherwise, if T≠P ⊗P T, then there would exist a non-singular Q∈Rn×n with
R
∈ α ∀ α ≠ I ;
Q n such that T=Q
(
P ⊗PT)
−1Q so that( )
(
⊕ −)
≠ ⊕ ⎜⎝⎛ − ⎟⎠⎞ −T A A
T 1
J J
T A A
T provided that
(
P ⊗P T)
−1(
A⊕( )
−AT)(
P ⊗P T)
=JA⊕ ⎝⎛ −⎜ JAT⎞⎟⎠Thus, note that:
(
)
(
⊕ −)
= = μ =σ ∑μ
=1 i ii 2
T n
A A card
( )
= μ = μ ≥νμ ≥ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛
μ ∑ ∑
∑ μ
= μ
= μ
= i 1
2 i 1
i ii 2
1
i i
0
( )
21 i j 1 ij 1
i 2 i 1
i ii
0 ⎟⎟
⎠ ⎞ ⎜⎜
⎝ ⎛
ν =
ν = ν = ν
≥ ∑ ∑ ∑ ∑μ
= μ = μ
= μ
=
( ) 2 ( ) n
2
1
i j i 1 ij 1
i j i 1 ij
≥ ν −
ν =
ν ∑ ∑
∑ ∑ μ
= μ
= ≠ μ
= μ
= ≠ −
Those results follow directly from the properties of the Kronecker sum A⊕Bof n- square real matrices A and
T
A
B=− since direct inspection leads to:
(1) 0∈ σ
(
A⊕( )
−AT)
with algebraic multiplicity( )
0 n1 i
2 i 1
i 2 i 1
i ii
≥ ν ≥ μ = μ ≥
μ ∑μ
= ∑
μ = ∑μ
= since there are at
least ∑ = μ n
1 i
2
i zeros in
(
( )
)
T A A⊕ −σ (i.e. the algebraic
multiplicity of 0∈σ
(
A⊕(
−AT)
)
is at least ∑ = μ n1 i
2 i ) and since νi≥1; ∀i∈n. Also, a simple computation of the number of eigenvalues of A⊕
(
−AT)
yields( )
(
)
21 i i 1
i ii 2
T n
A A card
⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
μ = μ = = −
⊕
σ ∑μ
= ∑μ
= .
(2) The number of linearly independent vectors in S is ∑μ
= ∑μ
= ∑μ
= ∑μ
= ∑
μ
= ⎟ ≥ ν = ν
⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
ν = ν =
ν
1 i
2 i 1
i ii 2
i 1 i 1
i j 1 ij
since the total number of Jordan blocks in the Jordan canonical form of A is ∑
μ =1ν
i i
.
(3) The number of Jordan blocks associated with
( )
(
A AT)
0∈σ ⊕ − in the Jordan canonical form of
( )
(
A⊕ − AT)
is ν
( )
= ∑μ ≤ν =1 i2 i v
0 , with νii=ν2ii ; n
i∈
∀ . Thus:
( )
(
)
∑μ= ∑μ
= μ = μ
= −
⊕ σ
1 i
2 i 1
i ii T
0 A A
card ,
(
)
(
)
∑
μ= μ − = −
⊕ σ
1 i
2 i 2
T
0 A A n
card
(
)
(
)
( )
∑
μ= − = ν − = − ⊕
1 i
2 i 2
2 T
v n
0 n A A
rank ,
( )
(
)
( )
∑μ= = ν = − ⊕
1 i
2 i
T 0 v
A A Ker dim
(4) There are at least ν
( )
0 linearly independent vectors in S:=span{
zi⊗x j,∀i,j∈n}
. Also, the total number of Jordan blocks in the Jordan canonical form of(
)
(
T)
A
A⊕ − is
( )
( ) ( )0
2 0 S
dim
1
i j i 1 ij 2
1 i i 1
i j 1 ij
ν ≥ ν +
ν = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛
ν = ⎟ ⎟ ⎠ ⎞ ⎜
⎜ ⎝ ⎛
ν =
=
ν ∑μ
= ∑
μ = ≠ ∑μ
= ∑μ
Property (i) has been proved. Property (ii) follows directly from the orthogonality in Rn2 of its range and null subspaces. Expressions which calculate the sets of matrices which commute and which do not commute with a given one are obtained in the subsequent result:
Theorem 2.4. The following properties hold:
(i) X∈CA iff
(
A⊕(
−AT)
)
v( )
X =0 ⇔ X∈CA iff( )
( )
( )
T2 T T T 2 T
X v , A A X v F X v
11 ⎟⎠ ⎞ ⎜
⎝ ⎛ −
= −
12
for any
( )
(
12)
1 11 21 22
2 Ker A A A A X
v ∈ − − , where
2 2 n
n F ,
E ∈R × are permutation matrices and X∈Rn×n and
( )
n2X
v ∈R are defined such that: (a) v
( )
X:=F−1v( )
X ,(
A)
A: E(
A(
A)
)
FA⊕ − T ≈ = ⊕ − T ; ∀X∈CA (2.3)
where
( )
( )
( )
2 T n2T 1 T
) X v , X v ( X
v = ∈R with
( )
( )0 1X
v ∈Rν and
( )
2 n ( )02
X
v ∈R −ν
(b) The matrix A11 ( ) ( )0 0
ν × ν
∈R is non-singular in the block matrix partition A:=Blockmatrix
(
Aij;i,j∈2)
with A12 ( )0 n2
× ν
∈R , 21
(
n ( )0)
( )02
A ∈R −ν ×ν and
( )
(
n 0)
(
n ( )0)
222 2
A ∈R −ν × −ν .
(ii) X∈CA, for any given A
( )
≠0 ∈Rn×n, iff(
)
(
A⊕ −AT)
v( ) ( )
X =v M (2.4) for some M( )
≠0 ∈Rn×nsuch that :(
)
(
A A)
rank(
A(
A)
, v( )
M)
n( )
0rank ⊕ − T = ⊕ − T = 2−ν
(2.5) Also,
(
)
(
)
( ) ( )
{
X : A A v X v M:
CA= ∈Rn×n ⊕ − T = for any
( )
n n0
M ≠ ∈R × satisfying
(
)
(
A A)
rank(
A(
A)
, v( )
M)
n( )
0}
rank ⊕ − T = ⊕ − T = 2−ν
(2.6) Also, with the same definitions of E , F and X in (i),
A C X∈ iff
( )
(
( )
( )
( )
)
T2 T T 1 1 T 2 T T 1 1 1 T
X v , A A X v A M v F X
v = − − −
12
(2.7) wherev
( )
X2 is any solution of the compatible algebraic system(
)
( ) ( )
( )
11 1 1 1 2 2 2
2 1 1 1 1 1 2 2
2 A A A v X v M A A v M
A − − = − −
(2.8) for some M
( )
≠0∈Rn×nsuch thatX, M∈Rn×n and are defined according to v( )
X =Fv( )
X and( )
n n0 M F M E
M= ≈ ≠ ∈R × and
( )
( )
(
( )
T( )
)
T2 T
1 M ,v M v
E M v E M
v = = .
Proof: First note from Proposition 2.1 that X∈CA iff
(
)
(
A⊕ −AT)
v( )
X =0 since( )
(
(
T)
)
A A
Ker X
v ∈ ⊕ − . Note also from Proposition
2.1 , that X∈CA iff v
( )
X ∈Im(
A⊕(
−AT)
)
.Thus, AC
X∈ iff v
( )
X is a solution to the algebraic compatible linear system:(
)
(
A⊕ −AT)
v( ) ( )
X =v M for any M( )
≠0 ∈Rn×nsuch that :(
)
(
A A)
rank(
A(
A)
, v( )
M)
n( )
0rank ⊕ − T = ⊕ − T = 2−ν
From Theorem 2.3, the nullity and the rank of A⊕
(
−AT)
are, respectively, dimKer(
A⊕(
−AT)
)
=ν( )
0(
)
(
A A)
n( )
0rank ⊕ − T = 2−ν . Therefore, there exist permutation matrices E,F∈Rn2×n2such that there exists an equivalence transformation:
( )
A A: E(
A( )
A)
F Blockmatrix(
A ;i,j 2)
A⊕ − T ≈ = ⊕ − T = ij ∈
such that A11 is square non-singular and of order ν0 . Define M=EMF≈M
( )
≠0∈Rn×n .Then, the linear algebraic systems(
A⊕(
−AT)
)
v( ) ( )
X =v M , and(
)
(
)
( )
( )
( )
( )
( )
⎥⎦ ⎤ ⎢ ⎣ ⎡ = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢
⎣ ⎡ = −
⊕
2 1 2
1 22
21 12 11 T
M v
M v X v
X v A A
A A X v F A A
E
( )
(
( )
( )
)
(
)
( ) ( )
( )
11 1 1 21 2 2
2 1 1 1 1 21 22
2 2 1 1 1 1 1 1
M v A A M v X V A A A A
X v A M v A X V
− −
−
− =
−
− =
⇔
(2.9) are identical if X and M are defined according to
( )
X Fv( )
Xv = and v
( )
M =Ev( )
M .As a result, Properties (i)-(ii) follow directly from (2.9) for M=M=0 and for any M satisfying :(
)
(
A A)
rank(
A(
A)
, v( )
M)
n( )
0rank ⊕ − T = ⊕ − T = 2−ν
, respectively.
III. PAIR-WISE COMMUTING MATRICES Consider the following sets:
(1) A set of nonzero p≥2 distinct pair-wise commuting matrices AC:=
{
Ai∈Rn×n:[
Ai,Aj]
=0; ∀i,j∈p}
(2) The set of matrices
[
]
{
X n n: X,Ai 0; Ai C}
: C M
C R A
A = ∈ × = ∀ ∈ which
commute with the set AC of pair-wise commuting matrices.
(3) A set of matrices
[
]
{
R A}
A:= X∈ n×n: X,Ai =0; ∀Ai∈
C which
commute with a given set of nonzero p matrices
{
A ; i p}
:= i∈Rn×n ∀ ∈
A which are not necessarily
The complementary sets of MCAC and CA are C
C
M A and CA , respectively, so that
C
C M B
n n
A
R × ∋ ∈ if
C
MC
B∉ A and
A
Rn×n ∋B∈C if B∉CA. Note that CAC =MCAC for a set of pair-wise commuting matrices AC so that the notation MCAC is directly referred to a set of matrices which commute with all those in a set of pair-wise commuting matrices. The following two basic results follow concerning commutation and non- commutation of two matrices:
Proposition 3.1. The following properties hold:
(i)
(
)
(
( )
)
( )
Ker A A ; i pA v
p i j
T j j
i ∈ ⊕ − ∀ ∈
⇔
∈ ≠I
v
(
A)
Ker(
A( )
A)
; i p pj 1 i
T j j
i ∈ ⊕ − ∀ ∈
⇔
≤ ≤ + I (ii) Define Ni
(
AC)
:=( )
(
)
(
)
(
)
[
]
Tp T p 1 i T
1 i 1 i T
1 i 1 T
1 A A A A A A A
A ⊕ − L − ⊕ − − + ⊕ − + L ⊕ − (p−1)n2×n2
∈R . Then Ai∈AC;∀i∈p iff
(
A)
KerN(
)
; i pv i ∈ i AC ∀ ∈
(iii)
( )
(
(
)
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩
⎪ ⎨ ⎧
∈ −
⊕ ∈
∈ =
∈ ×
I p
i i C
T i i n
n :v X Ker A A ;A
X : C M
C R A
A
( )
(
)
{
}
{ }
n nC C
n
n :v X KerN C 0
X
C
×
× ∈ ⊃ ⊃ ⊃ ∈
∈
= R A A A R
where
(
)
:=N AC
( ) ( )
(
)
[
A1T A1 AT2 A2 ApT Ap]
pn n ,Ai C2 2 T
A
R ∈
∈ −
⊕ −
⊕ −
⊕ L ×
(iv)
( )
(
(
)
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩
⎪ ⎨ ⎧
∈ −
⊕ ∈
∈ =
∈ ×
U
pi
C i T i i n
n :v X Im A A ;A
X : C M
C R A
A
{
( )
(
C)
}
nn :v X ImN
X∈R ∈ A
= ×
(v)
( )
(
(
)
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩
⎪ ⎨ ⎧
∈ −
⊕ ∈
∈ =
∈ ×
A R
A i
p i
T i i n
n
A ; A A Ker X
v : X
:
C I
=
{
X∈Rn×n:v( )
X ∈KerN( )
A}
where( )
:=N A
(
)
(
)
(
)
[
]
T pn2 n2p T p 2
T 2 1 T
1 A A A A A
A ⊕ − ⊕ − L ⊕ − ∈R ×
A
∈ i A , (vi)
( )
(
(
)
)
⎪⎭ ⎪ ⎬ ⎫
⎪⎩ ⎪ ⎨ ⎧
∈ −
⊕ ∈
∈ =
∈ ×
U
p ii T i i
n n
A ; A A
Im X
v : X
:
CA R A
=
{
X∈Rn×n: v( )
X ∈ImN( )
A}
Proof: (i)The first part of Property (i) follows directly from Proposition 2.1 since all the matrices of AC pair-wise commute and any arbitrary matrix commutes with itself ( thus j = i may be removed from the intersections of kernels of the
first double sense implication). The last part of Property (i) follows from the anti-symmetric property of the commutator
[
Ai,Aj] [
= Aj,Ai]
=0;∀Ai,Aj∈AC what impliesp i ; Ai∈A C ∀ ∈
( )
(
( )
)
i j Cp j 1 i
T j j
i Ker A A ; A ,A
A
v ∈ ⊕ − ∀ ∈A
⇔
≤ ≤ + I
(ii) It follows from its equivalence with Property (i) since
(
)
(
( )
)
( )
Ii p jT j j
C
i Ker A A
N Ker
∈
≠ ⊕ −
≡
A .
(iii) Property (iii) is similar to Property (i) for the whose set
C
MA of matrices which commute with the set AC so that it contains ACand, furthermore ,
(
)
I(
( )
)
p i
T i i
C Ker A A
N Ker
∈ ⊕ −
≡
A .
(iv) It follows from
( )
(
)
(
( )
)
U I
p
j j p j C
T j j T
j
j A Ker A A ;A
A Im
∈ ⊕ − = ∈ ⊕ − ∈A
and R n2∋0∈Ker
(
Aj⊕(
−ATj)
)
I Im(
Aj⊕(
−ATj)
)
but Rn×n∋X=0 commutes with any matrix in Rn×n so thatC
C 0 MC
C M
0 n n
n n
A
A R
R × ∋ ∈ ⇔ × ∋ ∉ for
any given A C.
(v)-(vi) are similar to (ii)-(iv) except that the members of A do not necessarily commute. Concerning Proposition 3.1 (v)-(vi), note that if X∈CA then X≠0 since Rn×n∋0∈CA . The following result is related to the rank defectiveness of the matrix N
(
AC)
and any of their sub-matrices since AC is a set of pair-wise commuting matrices:Proposition 3.2. The following properties hold:
(
)
(
)
(
( )
T)
j j
C i C
2 rankN rankN rank A A
n > A ≥ A ≥ ⊕ − ;
p j , i ;
Aj∈ C ∀ ∈
∀ A
and, equivalently, ( ) ( )
(
N N)
det(
N ( )N ( ))
det(
A(
A)
)
0 det T AC AC = Ti AC i AC = j⊕ − Tj =; ∀Aj∈AC;∀i,j∈n.
Proof: It is a direct consequence from Proposition 3.1 (i) –(ii) since the existence of nonzero pair-wise commuting matrices (all the members of A C) implies that the above matrices
(
)
(
)
( )
Tj j
C i
C ,N , A A
N A A ⊕ − are all rank defective
and have at least identical number of rows than that of columns. Therefore, the square matrices
(
) (
)
T(
C)
i(
C)
i C C
T N ,N N
N A A A A and
( )
T jj A
A ⊕ − are all singular. Results related to sufficient conditions for a set of matrices to
any arbitrary matrix consist of pair-wise commuting matrices. Any set of matrices obtained by linear combinations of one of the above sets consist also of pair-wise commuting matrices. Any matrix commutes with any of its matrix functions etc. In the following, we discuss a simple, although restrictive, sufficient condition for rank defectiveness of N
( )
A of some set A of p square real n- matrices which may be useful as a test to elucidate the existence of a nonzero n- square matrix which commutes with all matrices in this set. Another useful test obtained from the following result relies on a necessary condition to elucidate if the given set consists of pair-wise commuting matrices.Theorem 3.3: Consider any arbitrary set of nonzero n-square real matrices A:=
{
A1,A 2,...,Ap}
for any integer p≥1 and define matrices:( )
:= Ni A( )
(
)
(
)
(
)
[
]
Tp T p 1 i T
1 i 1 i T
1 i 1
T
1 A A A A A A A
A ⊕ − L − ⊕ − − + ⊕− + L ⊕ −
( )
[
( ) ( )(
)
]
Tp T p 2
T 2 1 T
1 A A A A A
A :
N A = ⊕ − ⊕ − L ⊕ −
Then, the following properties hold:
(i) rank
(
Ai ⊕(
−Ai)
)
≤rankNi( )
A ≤rankN( )
A <n2; pi∈
∀ .
(ii) Ker
(
A( )
A)
{ }
0 pi
T i
i⊕ − ≠
∈I so that:
( )
0 CAX ≠ ∈
∃ ,
( )
I(
(
)
)
p i
T i
i A
A Ker X
v C X
∈ ⊕ −
∈ ⇔ ∈ A
and
( )
U(
( )
)
p i
T i
i A
A Im X
v C X
∈ ⊕ −
∈ ⇔
∈ A
(iii) If A=AC is a set of pair-wise commuting matrices then
(
A)
Ker(
A( )
A)
; i pv
i \ p j
T j j
i ∈ ⊕ − ∀ ∈
∈I
(
A)
Ker(
A( )
A)
; i pv
p i
T i i
i ∈ ⊕ − ∀ ∈
⇔
∈I
(
)
(
( )
)
{ }
Ker A A ; i pA v
i \ p i
T i i
i ∈ ⊕ − ∀ ∈
⇔
∈ I (iv)
( )
(
(
)
)
⎭ ⎬ ⎫ ⎩
⎨ ⎧
∈ ∀ −
⊕ ∈
=
∈ ×
C i p
i
T i i n
n
C: X :v X Ker A A , A
MA R I A
{ }
n n C ∪ 0 ∈ ×⊃A R
with the above set inclusion being proper
Proof: (i) Any nonzero matrix Λ=diag
(
λ λ ...λ)
,( )
≠ ∈Rλ 0 is such that Λ
( )
≠0 ∈CAi(
∀i∈p)
so thatΛ∈CA . Thus,( )
KerN( )
A n rankN( )
Av
0≠ Λ∈ ⇔ 2>
( )
(
i(
i)
)
i rank A A
N
rank ≥ ⊕ −
≥ A ; ∀i∈p
and any given set A. Property (i) has been proved. (ii) The first part follows by contradiction. Assume
( )
(
A A)
{ }
0 Kerp i
T i
i⊕ − =
∈I then
( )
KerN( )
Av
0≠ Λ ∉ so that Λ=diag
(
λ λ ...λ)
∉CA, for any λ( )
≠0 ∈R what contradicts (i) . Also,( )
(
( )
T)
i i
A v X Ker A A
C X
i ⇔ ∈ ⊕ −
∈ ; ∀i∈p so that
( )
I(
( )
)
p i
T i
i A
A Ker X
v C X
∈ ⊕ −
∈ ⇔
∈ A what is
equivalent to its contrapositive logic proposition
( )
U(
( )
)
p i
T i
i A
A Im X
v C X
∈ ⊕ −
∈ ⇔
∈ A .
(iii) A=AC A C ; j
( )
i p, i p jA
i∈ ∀ ≠ ∈ ∀ ∈
⇔
A C ; j,i p j
A
i∈ ∀ ∈
⇔ since
i A i C
A ∈ ;∀i∈p
(
A)
Ker(
A( )
A)
; i pv
p i
T j j
i ∈ ⊕ − ∀ ∈
∈I
(
)
(
( )
)
{ }
Ker A A ; i pA v
i \ p i
T j j
i ∈ ⊕ − ∀ ∈
⇔
∈I On the other hand,
(
)
I(
(
)
)
i \ p j
T j j
i Ker A A
A v (
∈ ⊕ −
∈
( )
A C ; j pv
j A
i ∈ ∀ ∈
⇔ )foranyi
( )
<p ∈p. This assumption implies directly that:( )
A C ; j pv
j A
i ∈ ∀ ∈
(
)
I1 i
j A
1
i C j
A v
+ ∈
+ ∈
∧ foranyi
( )
<p ∈pwhich together with
(
)
I(
( )
)
1 i \ p j
T j j
1
i Ker A A
A v
+ ∈
+ ∈ ⊕ − implies that
(
A)
C ; j pv
j A 1
i+ ∈ ∀ ∈
(
A)
Ker(
A(
A)
)
) v1 i \ p j
T j j
1 i
(
I+ ∈
+ ∈ ⊕ −
⇒ for
( )
i+1 ∈pThus, it follows by complete induction that C
A
A=
(
)
(
( )
)
{ }
Ker A A ; i pA v
i \ p i
T j j
i ∈ ⊕ − ∀ ∈
⇔
∈ I and Property (iii) has been proved.
(iv) The definition of MAC follows from Property (iii) in order to guarantee that
[
X,Ai]
=0; ∀Ai∈A. The fact that such a set contains properly AC ∪{ }
0 followsdirectly from
(
)
(
MC)
{ }
0diag C
n n
C ≠ ∪ ∈
λ λ λ = Λ ∋
× A
R L A for
any R∋λ≠0.
Note that Theorem 3.3 (ii) extends Proposition 3.1 (v) since it is proved that CA\
{ }
0 ≠∅ because all nonzero(
)
ARn×n ∋Λ=diag λ λLλ∈C for any R∋λ≠0 and any set of matrices A. Note that Theorem 3.3 (iii) establishes
that
(
)
(
( )
)
{ }
Ker A A ; i pA v
i \ p i
T j j
i ∈ ⊕ − ∀ ∈
∈ I is a
necessary and sufficient condition for the set to be a set of commuting matrices A being simpler to test (by taking advantage of the symmetry property of the commutators)
(
A)
Ker(
A( )
A)
; i p v p i T j ji ∈ ⊕ − ∀ ∈
∈I . Further results
about pair-wise commuting matrices or the existence of nonzero commuting matrices with a given set are obtained in the subsequent result based on the Kronecker sum of relevant Jordan canonical forms:
Theorem 3.4. The following properties hold for any given set of n-square real matrices A=
{
A1,A2,...,Ap}
: (i) The set CA of matrices X∈Rn×nwhich commute with all matrices in A is defined by:( )
(
(
)
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ − ⊕ ∈ ∈ = = − − × I p 1 i T i 1 i T A A nn :vX Ker J J P P X : C i i R A (3.1) ( ) ( )
(
( )
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ ∈ ∀ − ⊕ ∈ ∧ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ ∈ ∈ = = − × I p 1 i T A A i i 1 i i nn :vX Im P P Y Y KerJ J ; i p
X i i R (3.2) ( ) ( )
(
(
( )
)
)
⎪⎭ ⎪ ⎬ ⎫ ⎪⎩ ⎪ ⎨ ⎧ − ⊕ ∈ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ ∈ ∈ = = = − ×Ip I
1 i p 1 i T A A 1 i i n n i i J J Ker Y , Y P P Im X v : X R (3.3) where Pi∈Rn×n is a non-singular transformation matrix such that Ai JA Pi1Ai Pi
i − = ∼ , i A
J being the Jordan canonical form of Ai.
(ii)
( )
{
}
(
(
(
T)
)
)
A A
p
i i i
J J Ker dim min C X : X v span
dim ∈ ≤ ⊕ −
∈ A
( )
⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ν = ν= ρ∑
= ∈ ∈ i 1 j 2 j i p i i p i min 0 min
min min
(
i( )
0)
p i 1 i 2 j i p i i μ ≤ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ μ ≤ ∈ ∑ ρ = ∈where νi(0) and νij are, respectively, the geometric multiplicities of 0∈σ
(
Ai⊕( )
−AiT)
and λij∈σ(
Ai)
and μi(0) and μij are, respectively, the algebraic multiplicities of 0∈σ(
Ai⊕( )
−ATi)
and λij∈σ(
Ai)
;i j∈ρ
∀ ( the number of distinct eigenvalues of i
A ),∀i∈p.
(iii) The set Aconsists of pair-wise commuting matrices,
namely CA=MCA , iff
( )
(
(
)
)
( )I
p 1 j i T i 1 i T A A
j Ker J J P P
A v i i = ≠ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ − ⊕ ∈ ; p j∈
∀ . Equivalent conditions follow from the second and third equivalent definitions of CAin Property (i).
Proof: If Ai JA Pi1AiPi i
− =
∼ , with JAi being the
Jordan canonical form of Ai then
( )
Ti A(
AT)
i1(
i( )
Ti)
ii A J J T A A T
A
i
i⊕ − = ⊕ −
∼ −
⊕ −
with Ti=Pi ⊗ PiT∈Rn2×Rn2 (see proof of Theorem
2.3) being non-singular ; ∀i∈p . Thus ,
( )
(
)
(
(
)
)
1i T A A i T i
i A T J J T
A i i − − ⊕ = −
⊕ so that:
( )
(
)
(
)
(
)
Tp T 2 T 1 T
1 A A2 A Ap A
A
N A =⎢⎣⎡ ⊕ − ⊕ − L ⊕ − ⎥⎦⎤
[
]
=TJTa ⎥⎦⎤ ⎢⎣
⎡ −
= T T
2 T 1 n
p U W W
I 2 (3.4)
where 2 2 n p n p p 2 1 T ...T T
Diag Block
:= ⎢⎣⎡ ⎥⎦⎤∈R ×
T ; 2 2 n n p T T p T 2 T
1 T T
T
:=⎢⎣⎡ − − − ⎥⎦⎤ ∈R ×
Ta L (3.5)
(
)
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⊕ − ⊕ = T A A T AA1 J 1 J p J p J Diag Block : L J 2 2 n p n p ×
∈R (3.6) Then,
( )
A Ker[
(
A(
A)
)
]
Ker(
JTa)
N Ker p 1 i T i
i ⊕ − =
= = I
(
)
(
)
(
)
[
]
(
)
I p 1 i T i 1 i T AA J P P
J Ker i i = − − ⊗ − ⊕ ≡
since T is non-singular. Thus,
( )
2 n DomX∈ A ⊂R
∀ :
( )
( )
AA v X KerN C
X∈ ⇔ ∈
( )
[
(
(
)
)
]
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⊕ ∈ ⇔ = I p 1 i T i i A A Ker X v( )
[
(
(
)
)
(
)
]
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⊗ − ⊕ ∈ ⇔ = − − I p 1 i T i 1 i T AA J P P
J Ker X v i i
( )
X Im(
(
P P)
(
Ker(
J(
J)
)
)
)
; i pv i i1 A AT
i
i⊕ − ∀ ∈
⊗ ∈
⇔ −
( )
Ip(
(
(
)
( )
)
)
( )
1 i
1 i
i P Y v X
P Im X v = − ⇔ ⊗ ∈ ⇔
(
)
( )
(
)
(
)
Ip 1 i i 1 ii P Y
P Im = − ⊗ ∈
where i
(
A(
AT)
)
i
i J
J Ker
Y ∈ ⊕ − ; ∀i∈p and
(
)
(
)
(
)
⎟⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − ⊕ ∈ = Ip 1 i T A Ai J i JKer
Y . Property (i) has been
proved. The first inequality of Property (ii) follows directly from Property (i). The results of equalities and inequalities in the second line of Property (ii) follow by the first inequality by taking into account and Theorem 2.3. Property (iii) follows from the proved equivalent definitions ofCAin Property (i) by taking into account that
[
A j, Aj]
=0;p j∈
∀ so that:
( )
I(
(
)
)
p 1 i T i 1 i T A A
j Ker J J P P
A v i i = − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ − ⊕ ∈
( )
(
(
)
)
( )I p 1 j i T i 1 i T A A
j Ker J J P P
A v i i = ≠ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎥⎦ ⎤ ⎢⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⊗ − ⊕ ∈ ⇔
; ∀j∈p
IV. FURTHER RESULTS AND SOME EXTENSIONS The extensions of the results for commutation of complex matrices is direct in several ways. It is first possible to decompose the commutator in its real and imaginary part and then apply the results of Sections 2-3 for real matrices to both parts as follows. Let A=Are+iAim and
m i e
r B
B
B= +i be complex matrices in Cn×n with Are, and Bre being their respective real parts , and Aim and
im
B , all in Rn×ntheir respective imaginary parts and 1
− =
i is the imaginary complex unity. Direct computations with the commutator of A and B yield:
[
A,B]
=(
[
Are,Bre] [
− Aim , Bim]
)
[
] [
]
(
Aim , B re + Are , Bim)
+i (4.1) The following three results are direct and allow to reduce the problem of commutation of a pair of complex matrices to the discussion of four real commutators:
Proposition 4.1. B∈CA⇔
[
] [
]
(
)
(
[
] [
]
)
(
Are,Bre = Aim, Bim ∧ Aim,Bre = Bim,Are)
.Proposition 4.2.
(
)
(
)
(
B re ∈ CAre ∩CAim ∧ B im ∈ CAim ∩CAre)
⇒ B∈CA Proposition 4.3.
(
)
(
)
(
A re ∈ CBre ∩CBim ∧ A im ∈ CBim ∩CBre)
⇒ B∈CA .
Proofs: Proposition 4.1 follow by inspection of (4.1). Proposition 4.2 implies that Proposition 4.1 holds with the four involved commutators being zero. Then the left condition of Proposition 4.2 implies that B∈CA, from Proposition 4.1, so that Proposition 4.2 holds. Proposition 4.3 is equivalent to Proposition 4.2.
Proposition 4.1 yields to the subsequent result Theorem 4.4. The following properties hold:
(i) Assume that the matrices A and Bre are given. Then, A
C
B∈ iff Bim satisfies the linear algebraic equation:
(
)
(
)
( )
(
(
)
)
( )
imT e r e r T m i m i e r T m i m i T e r e r B v A A A A B v A A A A ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ (4.2) for which a necessary condition is:
(
)
(
)
= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ T e r e r T m i m i A A A A rank(
)
(
)
(
(
)
)
( )
⎥⎥⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ − ⊕ − ⊕ − ⊕ − ⊕ e r T m i m i T e r e r T e r e r T m i m i B v A A A A A A A A rank (4.3)(ii) Assume that the matrices A and Bime are given. Then, A
C
B∈ iff Bre satisfies (4.2) for which a necessary condition is:
(
)
(
)
= ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ T m i m i T e r e r A A A A rank(
)
(
)
(
(
)
)
(
)
⎥⎥⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜ ⎝ ⎛ − ⊕ − ⊕ − ⊕ − ⊕ im T e r e r T m i m i T m i m i T e r e r B v A A A A A A A A rank(iii) Also, ∃B≠0 such that B∈CA with Bre=0 and 0
B≠
∃ such that B∈CA with Bim=0
Proof: (i) Eqn. 4.2 is a re-arrangement in an equivalent algebraic system of Proposition 4.1 in the unknown v
( )
Bim for given A andBre. The system is compatible if (4.2) holds from the Kronecker- Capelli theorem. The proof of Property (ii) is similar to that of (i) with the appropriate interchange of roles of Bre and Bim.(iii) Since
(
)
( )
2T e r e r T m i m i n A A A A rank < ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕
from Theorem 3.3
(i) then 0≠B=Bre∈CA iff
( )
AA A
e
r C C C
B
m i e
r ∩ ≠∅ ⊂
∈ . The same proof
follows for 0≠B=Bim∈CA since
( )
(
)
(
( )
)
2T e r e r T m i m i T m i m i T e r e r n A A A A rank A A A A rank < ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ = ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − ⊕ − ⊕ A more general result than Theorem 4.4 is the following: Theorem 4.5. The following properties hold:
(i) B∈CA∩Cn×n iff v
( )
B is a solution to the linear algebraic system :( )
(
)
( )
(
)
(
)
( )
( )
( )
0 B v B v A A A A A A A A m i e r T e r e r T m i m i T m i m i T e r e r = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⊕ − − ⊕ ⊕ − − ⊕ (4.4) Nonzero solutions B∈CA , satisfying( )
( )
( )
(
)
(
(
)
)
( )
( )
⎥⎥⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⊕ − − ⊕ ⊕ − − ⊕ ∈ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ T e r e r T m i m i T m i m i T e r e r m i e r A A A A A A A A Ker B v B v ,always exist since
( )
(
)
( )
(
)
(
)
( )
{ }
2 n 2 T e r e r T m i m i T m i m i T e r e r 0 A A A A A A A AKer ≠ ∈R
⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⊕ − − ⊕ ⊕ − − ⊕
, and equivalently, since
( )
(
)
( )
(
)
(
)
( )
T 2 e r e r T m i m i T m i m i T e r e r n 2 A A A A A A A A rank < ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⊕ − − ⊕ ⊕ − − ⊕ (4.5) (ii) Property (ii) is equivalent toA C
which has always nonzero solutions since
( )
(
A ⊕ −A*)
<n 2Proof: (i) It follows in the same way as that of Theorem 4.4 by rewriting the algebraic system (4.3) in the form (4.4) which has nonzero solutions if (4.5) holds. But (4.5) always holds since B=A∈CA∩ Cn×n is nonzero if A is
nonzero and if A=0∈Cn×n then n
n A
C =C × .
(ii) Direct calculations yield the equivalence of (4.4) with the separation into real and imaginary parts of the subsequent algebraic system:
(
A⊗In−In⊗A*)
v( )
B(
)
⎥⎦ ⎤ ⎢
⎣ ⎡
⎟ ⎠ ⎞ ⎜
⎝
⎛ −
⊗ − ⊗ +
= T
m i T
e r n n m i e
r A I I A A
A i i
( ) ( )
(
v Bre +iv Bim)
=0 which is always solvable with a nonzero solution (i.e.compatible) since rank
(
A⊗In −In⊗A*)
<n2(otherwise , A
(
≠0)
∈CA). The various results of Section 3 for a set of distinct complexmatrices to pair-wise commute and for characterizing the set of complex matrices which commute with those in a given set may be discussed by more general algebraic systems like the above one with four block matrices
(
)
(
)
(
)
(
)
(
)
( )
⎥ ⎥ ⎥ ⎥
⎦ ⎤
⎢ ⎢ ⎢ ⎢
⎣ ⎡
⊕ −
− ⊕
⊕ −
− ⊕
T e jr e
r 2 T
m i 2 m
ji
T m ji m
ji T
e r 2 e
jr
A j
A A
A
A A
A A
for each j∈pin
the whole algebraic system. Theorem 4.5 extends directly for sets of complex matrices commuting with a given one and complex matrices commuting with a set of commuting complex matrices as follows:
Theorem 4.6. The following properties hold:
(i) Consider the sets of nonzero distinct complex matrices
{
A :i p}
:= i∈Cn×n ∈
A and
⎭ ⎬ ⎫ ⎩
⎨ ⎧
∈ ∀ ∈ =
⎥⎦ ⎤ ⎢⎣
⎡ ∈
= X × : X ,A 0;A , i p
:
CA Cn n i i A for
2
p≥ . Thus, CA∋X=Xre+iXre iff
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
0X v
X v
A A
A A
A A
A A
A A
A A
A A
A A
A A
A A
A A
A A
m i
e r
T e pr e
pr T
m pi m
pi
T m pi m
pi T
e pr e
pr
T e r 2 e
r 2 T
m i 2 m
i 2
T m i 2 m
i 2 T
e r 2 e
r 2
T e r 1 e
r 1 T
m i 1 m
i 1
T m i 1 m
i 1 T
e r 1 e
r 1
= ⎥ ⎥ ⎦ ⎤ ⎢
⎢ ⎣ ⎡
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦ ⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣ ⎡
⊕ −
− ⊕
⊕ −
− ⊕
⊕ −
− ⊕
⊕ −
− ⊕
⊕ −
− ⊕
⊕ −
− ⊕
M
(4.7) and a nonzero solutionX∈CAexists since the rank of the coefficient matrix of (4.7) is less than 2n2.
(ii) Consider the sets of nonzero distinct commuting
complex matrices AC:=
{
Ai∈Cn×n:i∈p}
and⎭ ⎬ ⎫ ⎩
⎨ ⎧
∈ ∀ ∈ =
⎥⎦ ⎤ ⎢⎣
⎡ ∈
= X × : X ,A 0; A , i p
: C
M A Cn n i i A for
2
p≥ . Thus, MCA∋X=Xre+iXre iff v (Xre) and )
(X
v im are solutions to (4.7).
(iii) Properties (i) and (ii) are equivalently formulated by from the algebraic set of complex equations:
( )X 0 v A A A
A A
A1* 1 *2 2 *p p ⎥⎦⎤* =
⎢⎣
⎡ ⎟
⎠ ⎞ ⎜ ⎝ ⎛− ⊕ ⎟
⎠ ⎞ ⎜ ⎝ ⎛ − ⊕ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −
⊕ L
(4.8) Outline of Proof: (i) It is a direct extension of Theorem 4.5 by decomposing the involved complex matrices in their real and imaginary parts since from Theorem 3.3 (i) both left block matrices in the coefficient matrix of (4.7) have rank less than n2. As a result, such a coefficient matrix has rank less than 2 n2so that nonzero solutions exists to the algebraically compatible system of linear equations (4.7) . As a result, a nonzero n-square complex commuting matrix exists.
(ii) It is close to that of (i) but the rank condition for compatibility of the algebraic system is not needed since the coefficient matrix of (4.7) is rank defective since
( ) (
)
(
)
Tm i j T re j T C
j v A ,v A
A ∈A ⇔ is in the null
space of the coefficient matrix; ∀j∈p.
(iii) Its proof is close to that of Theorem 4.5 (ii) and it is then omitted.
Remark 4.7. Note that all the proved results of Sections 2- 3 are directly extendable for complex commuting matrices, by simple replacements of transposes by conjugate transposes, without requiring a separate decomposition in real and imaginary parts as discussed in Theorem 4.5(ii) and Theorem 4.6 (iii).
Let f:C→C be an analytic function in an open set
( )
A σ ⊃D for some matrix A∈Cn×n and let p
( )
λ a polynomial fulfilling p( )
i( )
λk =f( )
i( )
λk ; ∀k∈σ( )
A ,{ }
0 1 m i∈ k − ∪∀ ; ∀k∈μ(the number of distinct elements in σ
( )
A , where mk is the index of λk , that is its multiplicity in the minimal polynomial of A. Then, f (A) is a function of a matrix A if f(A)=p(A), [8]. Some results follow concerning the commutators of functions of matrices. Theorem 4.8. Consider a nonzero matrixn n A C
B∈ ∩C × for any given nonzero A∈Cn×n. Then,
( )
n nA C B
f ∈ ∩ C × , and equivalently
( )
(
)
(
( )
*)
A A Ker B f
v ∈ ⊕ − , for any function
n n n n :
f C × →C × of the matrix B. Proof: For any B∈CA∩Cn×n:
[
A,B]
=0⇒(
λIn −B)
A=A(
λIn −B)
;∀λ∈C(
λIn −B)
1A=A(
λIn −B)
1;∀λ∈ ∩σ( )
B⇒ − − C (4.9)
( )
[
]
( )(
)
⎥⎦ ⎤ ⎢
⎣ ⎡
λ −
λ λ ∫ π =
⇒ f I B − d
2 1 A B f ,
A C n 1