Linear maps preserving idempotents of tensor products of matrices

JID:LAA AID:12569 /FLA [m1L; v 1.128; Prn:17/02/2014; 9:36] P.1 (1-15)Linear Algebra and its Applications ••• (••••) •••–•••

Contents lists available at ScienceDirect

Linear Algebra and its Applications

www.elsevier.com/locate/laa

Linear maps preserving idempotents of tensorproducts of matrices

Baodong Zheng a, Jinli Xu a,∗, Ajda Fošner b

a Department of Mathematics, Harbin Institute of Technology, Harbin 150001,PR Chinab Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper,Slovenia

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 September 2013Accepted 27 January 2014Available online xxxxSubmitted by R. Brualdi

MSC:15A6915A86

Keywords:Linear preserverTensor productIdempotent

Let F be a field of characteristic not 2 and Mn the algebraof all n × n matrices over F. The aim of this paper is tocharacterize linear maps φ : Mm1···ml → Mm1···ml such thatφ(A1 ⊗ · · · ⊗ Al) is an idempotent whenever A1 ⊗ · · · ⊗ Al isan idempotent.

© 2014 Elsevier Inc. All rights reserved.

1. Introduction and the Main Theorem

Let F be a field of characteristic not 2, Mn the algebra of all n× n matrices over F,and Pn the subset of Mn containing all idempotent matrices, i.e.,

Pn :={P ∈ Mn: P 2 = P

}.

* Corresponding author.E-mail addresses: [email protected] (B. Zheng), [email protected] (J. Xu), [email protected]

(A. Fošner).

http://dx.doi.org/10.1016/j.laa.2014.01.0360024-3795/© 2014 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.laa.2014.01.036

http://www.ScienceDirect.com/

http://www.elsevier.com/locate/laa

mailto:[email protected]



http://dx.doi.org/10.1016/j.laa.2014.01.036

JID:LAA AID:12569 /FLA [m1L; v 1.128; Prn:17/02/2014; 9:36] P.2 (1-15)2 B. Zheng et al. / Linear Algebra and its Applications ••• (••••) •••–•••

Suppose that m,n � 2 are positive integers. Then for A ∈ Mm and B ∈ Mn, wedenote by A ⊗ B ∈ Mmn their tensor (Kronecker) product. In many pure and appliedstudies, one considers the tensor product of matrices (see, for example, [1,13,14,19]).Most noticeably, the tensor product is often used in quantum information science [17].In a quantum system, quantum states are represented as density matrices, i.e., positivesemi-definite matrices with trace one. If A ∈ Mm and B ∈ Mn are two quantum statesin two quantum systems, then their tensor product A ⊗ B describes the joint state inthe bipartite system in which the general states are density matrices in Mmn. Of course,a general observable on the bipartite system corresponds to Hermitian C ∈ Mmn, andthe set of observables of the form A⊗B with Hermitian A ∈ Mm, B ∈ Mn is a very small(measure zero) set. Nevertheless, one may be able to extract useful information aboutthe bipartite system by focusing on the set of tensor product of matrices. In particular,in the study of linear operators φ : Mmn → Mmn on bipartite systems, the structureof φ can be determined by studying φ(A ⊗ B) with A ∈ Mm, B ∈ Mn (see [6,9,12]and references therein). More generally, one may also consider tensor states and generalstates in a multipartite system Mm1 ⊗ · · · ⊗Mml

= Mm1···ml.

The purpose of this paper is to determine the structure of linear maps on multipartitesystems which preserve idempotents of tensor products of matrices. Let us point out thatour study is mainly theoretical and somehow remotely related to quantum informationscience mentioned above. The real motivation for our work comes from linear preserverproblems. Our study in this paper is more related to the study in [12], in which the authorcharacterizes operations preserving separability measures via linear preserver problems.In the last few decades there has been considerable interest in studying linear preserverson matrix algebras as well as on more general rings and operator algebras. For example,Frobenius [10] showed that a linear operator φ : Mn → Mn satisfies

det(φ(A)

)= det(A), ∀A ∈ Mn,

if and only if there are U, V ∈ Mn with det(UV ) = 1 such that φ has the form

A �→ UAV or A �→ UAtV,

where At denotes the transpose of A. Clearly, a map of the above form is linear andleaves the determinant function invariant. But it is quite surprising that a linear mapwhich preserves the determinant function must be of this form. Furthermore, in [4]Dieudonné showed that a bijective linear operator φ : Mn → Mn maps the set ofsingular matrices into itself if and only if there are invertible U, V ∈ Mn such that φ hasthe above (standard) form. For results on linear preserver problems one may see [15] andreferences therein and for more new results on preserver problems we refer the reader to[2,16,20].

In connection to the linear preservers of idempotents, Chan and Lim [3] studied linearmaps on Mn which preserve k-potent matrices for k = 2, 3. In particular, they proved

JID:LAA AID:12569 /FLA [m1L; v 1.128; Prn:17/02/2014; 9:36] P.3 (1-15)B. Zheng et al. / Linear Algebra and its Applications ••• (••••) •••–••• 3

that a linear map φ : Mn → Mn maps idempotent matrices into idempotent matricesif and only if there exists an invertible Q ∈ Mn such that either

φ(A) = QAQ−1, A ∈ Mn,

or

φ(A) = QAtQ−1, A ∈ Mn.

Continuing this line of investigations, we study linear idempotent preservers on multi-partite systems.

Before writing our Main Theorem, let us introduce basic definitions and fix somenotation. First of all, throughout the paper, l, n and m1, . . . ,ml � 2 are positive integerswith n � m1 · · ·ml. We say that a linear map φ : Mm1···ml

→ Mn preserves idempotentsof tensor products of matrices if for any A1 ⊗ · · · ⊗ Al ∈ Mm1···ml

, the image φ(A1 ⊗· · · ⊗ Al) is an idempotent whenever A1 ⊗ · · · ⊗ Al is an idempotent. We call a linearmap π on Mm1···ml

canonical, if π maps A1 ⊗ · · · ⊗ Al to τ1(A1) ⊗ · · · ⊗ τl(Al), whereτk : Mml

→ Mmlis either the identity map X �→ X or the transposition map X �→ Xt,

k = 1, . . . , l.Our main result reads as follows.

Main Theorem. A linear map φ : Mm1···ml→ Mn preserves idempotents of tensor

products of matrices if and only if either φ = 0 or n = m1 · · ·ml and there is aninvertible matrix Q ∈ Mn and a canonical map π on Mm1···ml

such that φ has the form

φ(A1 ⊗ · · · ⊗Al) = Qπ(A1 ⊗ · · · ⊗Al)Q−1 (1)

for all Ak ∈ Mmk, k = 1, . . . , l.

Remark 1.1. In most of the previous studies of linear preservers involving tensor productspaces, one imposes the assumption that the preservers send tensor states to tensorstates. As a result, the preservers have the form

A1 ⊗A2 ⊗ · · · ⊗Al �→ τσ(1)(Aσ(1)) ⊗ τσ(2)(Aσ(2)) ⊗ · · · ⊗ τσ(l)(Aσ(l))

for some permutation σ ∈ Sl. In our case, we do not assume that the preservers sendtensor states to tensor states. Nevertheless, our result shows that, up to a similarity, westill have the above form. Let us also point out that this paper is not the first one dealingwith such a challenging problem. Recently, there has been a series of papers studyingpreservers on tensor products of matrices without requiring the maps sending tensormatrices to tensor matrices (see, for example, [5–8] and references therein).


In the following, Ik denotes the k×k identity matrix and E(k)ij , 1 � i, j � k, the k×k

matrix whose all entries are equal to zero except for the (i, j)-th entry which is equalto 1.

Remark 1.2. In our Main Theorem we characterize maps which preserve idempotents oftensor products of matrices and that the resulting map may not preserve all idempotentsof Mm1···ml

. More precisely, if l > 1, then the map φ of the form (1) may not satisfyφ(Pm1···ml

) ⊆ Pn. For example, let

φ(A1 ⊗A2 ⊗ · · · ⊗Al) = At1 ⊗A2 ⊗ · · · ⊗Al

for all A1 ⊗A2 ⊗ · · · ⊗Al ∈ Mm1···ml. Denote

X := E(m1)11 ⊗ E

(m2)11 + E

(m1)12 ⊗ E

(m2)12 + E

(m1)21 ⊗ E

(m2)21 + E

(m1)22 ⊗ E

(m2)22

and

Y := E(m1)11 ⊗ E

(m2)11 + E

(m1)21 ⊗ E

(m2)12 + E

(m1)12 ⊗ E

(m2)21 + E

(m1)22 ⊗ E

(m2)22 .

Of course, X,Y ∈ Mm1m2 . Now, let

C := 12X ⊗ Im3···ml

∈ Mm1···ml.

Then C ∈ Pm1···mland

φ(C) = 12Y ⊗ Im3···ml

/∈ Pm1···ml.

In a special case, when l = 2 and m1 = m2 = 2, we have

C =

⎡⎢⎢⎣

12 0 0 1

20 0 0 00 0 0 012 0 0 1

2

⎤⎥⎥⎦ ∈ P4 and φ(C) =

⎡⎢⎢⎣

12 0 0 00 0 1

2 00 1

2 0 00 0 0 1

2

⎤⎥⎥⎦ /∈ P4.

Remark 1.3. In the paper, we deal just with the case n � m1 · · ·ml. Here, it wouldbe also interesting to study linear maps φ : Mm1···ml

→ Mn with n > m1 · · ·ml whichpreserve idempotents of tensor products of matrices. The structure of these kind of mapscan be very wild and even for l = 2 this is still an open problem.

2. Preliminary results

Before proving our main result, let us write four simple lemmas which we will needin the sequel. We start with a well-known exercise which is taken from the book [11].


Lemma 2.1. (See [11, p. 62, Exercise 1].) Let k be a positive integer and suppose thatP1, . . . , Pk ∈ Pn are idempotents such that Pi + Pj ∈ Pn for all 1 � i < j � k. Denoteri = rank(Pi), i = 1, . . . , k. Then there exists an invertible matrix T ∈ Mn such that

Pi = T Diag(0, . . . , 0, 1, . . . , 1︸︷︷︸ri

, 0, . . . , 0)T−1, i = 1, . . . , k,

where Diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0) is the diagonal matrix in which all diagonal entriesare zero except those in (r1 + · · · + (ri−1 + 1))-th to (r1 + · · · + ri)-th rows.

In the following, 0k will denote the k × k zero matrix.

Lemma 2.2. Let r, s be positive integers with r + s � n. If P ∈ Pr and X, 0s ⊕ P ⊕0n−r−s −X ∈ Pn, then there exists Xr ∈ Pr such that X = 0s ⊕Xr ⊕ 0n−r−s.

Proof. Since P ∈ Pr, we can write P = T (It ⊕ 0r−t)T−1 for some invertible T ∈ Mr

and some integer t � r. Set⎡⎣ Is 0 0

0 T−1 00 0 In−r−s

⎤⎦X

⎡⎣ Is 0 0

0 T 00 0 In−r−s

⎤⎦ =

⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦ ,

where X11 ∈ Ms, X22 ∈ Mt, and X33 ∈ Mn−s−t. Obviously,⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦ ∈ Pn

and⎡⎣ 0s 0 0

0 It 00 0 0n−s−t

⎤⎦−

⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦ ∈ Pn.

Hence,⎡⎣ 0s 0 0

0 It 00 0 0n−s−t

⎤⎦⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦ +

⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦⎡⎣ 0s 0 0

0 It 00 0 0n−s−t

⎤⎦

= 2

⎡⎣X11 X12 X13X21 X22 X23X31 X32 X33

⎤⎦ .

A straightforward computation shows that Xij = 0, i, j = 1, 2, 3, with one possibleexception X22. Thus, X = 0s ⊕ T (X22 ⊕ 0r−t)T−1 ⊕ 0n−r−s, as desired. �


Recall that if a matrix U ∈ Mkh commutes with Ik⊗S for all real symmetric S ∈ Mh,then U has the form W ⊗ Ih with W ∈ Mk. This simple observation will be used in theproof of our next lemma.

Lemma 2.3. Let π1, π2 be canonical maps on Mm1···mland U, V ∈ Mm1···ml

invertiblematrices. If π1(X)U = V π2(X) for all X ∈ Mm1···ml

, then π1 = π2 and U = V =λIm1···ml

for some nonzero scalar λ ∈ F.

Proof. Since π1, π2 are canonical maps on Mm1···ml, we can write

π1 = τ1 ⊗ · · · ⊗ τl and π2 = η1 ⊗ · · · ⊗ ηl,

where each of the maps τk and ηk, k = 1, . . . , l, is either the identity map X �→ X or thetransposition map X �→ Xt.

Clearly, πi(Im1···ml) = Im1···ml

, i = 1, 2. Hence, U = V . To prove that U is a scalarmultiple of the identity map, we use the induction on l. The case l = 1 is obvious. So, letl � 2 and assume that the statement holds true for l−1. Note that for any real symmetricS ∈ Mml

, we have πi(Im1···ml−1 ⊗ S) = Im1···ml−1 ⊗ S, i = 1, 2. This implies that U

commutes with Im1···ml−1 ⊗ S for all real symmetric S ∈ Mml, and, thus, U = W ⊗ Iml

for some matrix W ∈ Mm1···ml−1 . Since U is invertible, W has to be invertible as well.Let us define linear maps

π1 = τ1 ⊗ · · · ⊗ τl−1 and π2 = η1 ⊗ · · · ⊗ ηl−1.

It is clear that π1 and π2 are canonical maps on Mm1···ml−1 . Let Y ∈ Mm1···ml−1 be anarbitrary matrix. Then

(π1(Y )W

)⊗ Iml

= π1(Y ⊗ Iml)U = Uπ2(Y ⊗ Iml

) =(Wπ2(Y )

)⊗ Iml

and, thus, π1(Y )W = Wπ2(Y ). Consequently, by the induction hypothesis, W =λIm1···ml−1 for some nonzero scalar λ. Therefore, U = λIm1···ml

and

π1(X) = 1λπ1(X)U = 1

λUπ2(X) = π2(X)

for all X ∈ Mm1···ml. �

Lemma 2.4. Let π1, π2 be canonical maps on Mm1···mland U ∈ Mm1···ml

. If π1(P )U =π1(P )Uπ2(P ) for all P = P1 ⊗ · · · ⊗ Pl ∈ Pm1···ml

, then U = λIm1···mlfor some scalar

λ ∈ F. Moreover, if U is invertible, then π1 = π2 and λ = 0.

Proof. As above, let us write

π1 = τ1 ⊗ · · · ⊗ τl and π2 = η1 ⊗ · · · ⊗ ηl,


where each of the maps τk and ηk, k = 1, . . . , l, is either the identity map X �→ X or thetransposition map X �→ Xt. Now, let us define linear maps

π1 = τ2 ⊗ · · · ⊗ τl and π2 = η2 ⊗ · · · ⊗ ηl.

Obviously, π1 and π2 are canonical maps on Mm2···ml.

Set

Pi = E(m1)ii ⊗ Im2···ml

, i = 1, . . . ,m1,

and

Qij = 12(E

(m1)ii + E

(m1)jj + E

(m1)ij + E

(m1)ji

)⊗ Im2···ml

, 1 � i < j � m1.

Clearly, πk(Pi) = Pi, πk(Qij) = Qij and, thus,

PiU = PiUPi and QijU = QijUQij .

From the first equality, it follows that U = Diag(U1, . . . , Um1) with Uk ∈ Mm2···ml,

k = 1, . . . ,m1, and, from the second equality, we get Uk = U1, k = 2, . . . ,m1. DenoteU = U1. Then

U = Diag(U , . . . , U) = Im1 ⊗ U .

For any P2 ⊗ · · · ⊗ Pl ∈ Pm2···ml, we have

π1(Im1 ⊗ P2 ⊗ · · · ⊗ Pl)U = π1(Im1 ⊗ P2 ⊗ · · · ⊗ Pl)Uπ2(Im1 ⊗ P2 ⊗ · · · ⊗ Pl).

This yields that

(Im1 ⊗ π1(P2 ⊗ · · · ⊗ Pl)

)(Im1 ⊗ U)

=(Im1 ⊗ π1(P2 ⊗ · · · ⊗ Pl)

)(Im1 ⊗ U)

(Im1 ⊗ π2(P2 ⊗ · · · ⊗ Pl)

).

Hence, π1(P )U = π1(P )U π2(P ) for all P = P2 ⊗ · · · ⊗ Pl ∈ Pm2···ml. From this, one can

deduce that there exists λ ∈ F such that U = λIm1···ml.

At the end, suppose that U is invertible, i.e., U = λIm1···mlfor some nonzero scalar λ.

Let P = (E(m1)11 + E

(m1)12 ) ⊗ Im2···ml

and assume that τ1 = η1, i.e., τ1 is the identitymap and η1 is the transposition or vice versa. If the first possibility holds, then π1(P ) =(E(m1)

11 + E(m1)12 ) ⊗ Im2···ml

and π2(P ) = (E(m1)11 + E

(m1)21 ) ⊗ Im2···ml

. Therefore,

π1(P )U = λ(E

(m1)11 + E

(m1)12

)⊗ Im2···ml

and


π1(P )Uπ2(P ) = λ((E

(m1)11 + E

(m1)12

)⊗ Im2···ml

)((E

(m1)11 + E

(m1)21

)⊗ Im2···ml

)= λ

(E

(m1)11 + E

(m1)12

)⊗ Im2···ml

,

a contradiction. In a similar way we get a contradiction if τ1 is the transposition andη1 is the identity map. So, τ1 = η1. Now, if k = 2, . . . , l, we set P = Im1···mk−1 ⊗(E(mk)

11 + E(mk)12 ) ⊗ Imk+1···ml

and, using the same arguments as above, we see thatτk = ηk. �3. Proof of the Main Theorem

Since the sufficiency part of the Main Theorem is clear, we consider only the necessitypart. So, throughout this section, we assume that φ : Mm1···ml

→ Mn is a linear mapwhich preserves idempotents of tensor products of matrices. Let us start with some basicfacts.

Claim 1. If P ∈ Pm1 and

φ(P ⊗ Im2···ml) = 0s ⊕ Ir ⊕ 0n−r−s

for some nonnegative integers r and s, then there exists a linear map ψ : Mm2···ml→ Mr

such that

φ(P ⊗X) = 0s ⊕ ψ(X) ⊕ 0n−r−s

for all X ∈ Mm2···ml. Moreover, ψ preserves idempotents of tensor products of matrices.

Proof. Let Γl−1 = {P2 ⊗ · · · ⊗Pml: Pi ∈ Pmi

, i = 2, . . . , l}. Note that the set Γl−1 con-tains all idempotents in Mm2···ml

which can be written as a tensor product of matrices.Thus, the linear span of the set Γl−1 is equal to Mm2···ml

. Since φ is linear, it sufficesto prove that for any X ∈ Γl−1, the image of P ⊗X has the form

⎡⎣ 0s 0 0

0 Xr 00 0 0n−r−s

⎤⎦ (2)

for some Xr ∈ Pr. Namely, this yields that for all X ∈ Mm2···mlthe image of P ⊗X has

the above form with Xr ∈ Mr. In other words, there exists a map ψ : Mm2···ml→ Mr

such that

φ(P ⊗X) = 0s ⊕ ψ(X) ⊕ 0n−r−s, X ∈ Mm2···ml.

Furthermore, by the property of φ, we can see that ψ is linear and it preserves idempo-tents of tensor products of matrices, as desired.


Let us first define a chain of sets

Γ0 = {Im2 ⊗ · · · ⊗ Iml},

Γ1 = {P2 ⊗ Im3···ml: P2 ∈ Pm2},

Γ2 = {P2 ⊗ P3 ⊗ Im4···ml: Pi ∈ Pmi

, i = 2, 3},...

Γl−1 = {P2 ⊗ · · · ⊗ Pml: Pi ∈ Pmi

, i = 2, . . . , l}.

We use the induction on k = 0, 1, . . . , l−1 to prove that φ(P ⊗X) has the form (2) for allX ∈ Γk. Note the case k = 0 is just the assumption φ(P ⊗ Im2···ml

) = 0s ⊕ Ir ⊕ 0n−r−s.So, let k � 1 and assume that our statement holds true for k − 1.

Set

X = P2 ⊗ · · · ⊗ Pk ⊗ Pk+1 ⊗ Imk+2···ml∈ Γk,

and

Y = P2 ⊗ · · · ⊗ Pk ⊗ Imk+1 ⊗ Imk+2···ml∈ Γk−1

for some idempotents Pi ∈ Pmi, i = 2, . . . , k + 1. Then

Y −X = P2 ⊗ · · · ⊗ Pk ⊗ (Imk+1 − Pk+1) ⊗ Imk+2···ml∈ Γk.

Of course, the matrices φ(P⊗X), φ(P⊗Y ), φ(P⊗Y )−φ(P⊗X) are all n×n idempotentsand, by the induction hypothesis, we have

φ(P ⊗ Y ) =

⎡⎣ 0s 0 0

0 Yr 00 0 0n−r−s

⎤⎦

for some idempotent Yr ∈ Pr. Using Lemma 2.2, we conclude that φ(P ⊗ X) has theform (2), as desired. �Claim 2. If m1 · · ·ml > n, then φ = 0.

Proof. We use the induction on l. The case l = 1 is just Theorem 3.1 in [18]. Now,let l � 2 and assume that the statement holds true for l − 1. Since E

(m1)ii ⊗ Im2···ml

,i = 1, . . . ,m1, and (E(m1)

ii + E(m1)jj ) ⊗ Im2···ml

, 1 � i < j � m1, are all idempotents, weobtain, using the property of φ, that

φ(E


)∈ Pn, i = 1, . . . ,m1,


and

φ(E


)+ φ

(E

(m1)jj ⊗ Im2···ml

)∈ Pn, 1 � i < j � m1.

Denote

ri = rank(φ(E


)), i = 1, . . . ,m1.

Applying Lemma 2.1 and composing φ by the appropriate similarity transformation, wemay assume that

φ(E


)= Diag(0, . . . , 0, 1, . . . , 1︸︷︷︸

ri

, 0, . . . , 0),

where Diag(0, . . . , 0, 1, . . . , 1, 0, . . . , 0) is the diagonal matrix in which all diagonal entriesare zero except those in (r1 + · · · + (ri−1 + 1))-th to (r1 + · · · + ri)-th rows.

Since n < m1 · · ·ml we can find 1 � i � m1 such that ri < m2 · · ·ml. Forthe sake of the simplicity, let us assume for a moment that r1 < m2 · · ·ml. Thenφ(E(m1)

11 ⊗ Im2···ml) = Ir1 ⊕ 0n−r1 . Applying Claim 1, we see that there is a linear

map ψ : Mm2···ml→ Mr1 , which preserves idempotents of tensor products of matrices,

such that φ(E(m1)11 ⊗ X) = ψ(X) ⊕ 0n−r1 for all X ∈ Mm2···ml

. Using the inductionhypothesis, we conclude that ψ = 0. Therefore,

φ(E

(m1)11 ⊗X

)= 0, X ∈ Mm2···ml

. (3)

Note that if P = P2 ⊗ · · · ⊗ Pl ∈ Pm2···mlis an idempotent and λ ∈ F, then (E(m1)

11 +λE

(m1)1j ) ⊗ P , j = 2, . . . ,m1, is an idempotent as well. Thus, using the property of φ

and (3), we conclude that λφ(E(m1)1j ⊗ P ) ∈ Pn and, hence,

φ(E

(m1)1j ⊗ P

)= 0, j = 1, . . . ,m1.

Similarly,

φ(E

(m1)i1 ⊗ P

)= 0, i = 1, . . . ,m1.

Furthermore, if P = P2 ⊗ · · · ⊗ Pl ∈ Pm2···mlis an idempotent and 2 � i � l, then

( 12E

(m1)11 + 1

2E(m1)1j + 1

2E(m1)jj + 1

2E(m1)ii ) ⊗ P is an idempotent as well. According to the

property of φ and previous observations, we obtain

φ(E

(m1)ii ⊗ P

)= 0, i = 1, . . . ,m1.

Now, using similar arguments as above, we conclude that


φ(E

(m1)ij ⊗ P

)= 0, 1 � i, j � m1.

This yields that φ = 0, as desired. �In the following, we assume that n = m1 · · ·ml and φ = 0.

Claim 3. Suppose that F1, . . . , Fm1 are rank-one idempotents with Fi + Fj ∈ Pm1 , 1 �i < j � m1. If l � 2, then there exist an invertible matrix Q ∈ Mm1···ml

and linear mapsψk : Mm2···ml

→ Mm2···ml, k = 1, . . . ,m1, such that

φ(Fk ⊗X) = Q(E

(m1)kk ⊗ ψk(X)

)Q−1

for all X ∈ Mm2···mland k = 1, . . . ,m1. Moreover, maps ψk preserve idempotents of

tensor products of matrices.

Proof. Using a similar approach as in the proof of Claim 2, we show that

rank(φ(Fk ⊗ Im2···ml

))

= m2 · · ·ml, k = 1, . . . ,m1.

Namely, if rank(φ(Fk ⊗ Im2···ml)) > m2 · · ·ml for some 1 � k � m1, then there must

exists 1 � i � m1 such that rank(φ(Fi ⊗ Im2···ml)) < m2 · · ·ml. As above, this implies

that φ = 0, a contradiction. Thus, by Lemma 2.1, one can see that there exists aninvertible matrix Q ∈ Mm1···ml

such that

φ(Fk ⊗ Im2···ml) = Q

(E

(m1)kk ⊗ Im2···ml

)Q−1, k = 1, . . . ,m1.

We end the proof by applying Claim 1. �Now we are in the position to prove the necessity part of our Main Theorem. We use

the induction on l. The case l = 1 is just Theorem 3 in [3]. So, assume that l � 2 andthat the statement holds true for l− 1. Applying Claim 3 and induction hypothesis andcomposing φ by the appropriate similarity transformation, we may assume that

φ(E

(m1)kk ⊗X

)= E

(m1)kk ⊗ πk(X) (4)

for all X ∈ Mm2···mland k = 1, . . . ,m1, where πk are canonical maps on Mm2···ml

. Wedivide the rest of the proof into three steps.

Step 1. We claim that πk = π1 for all 2 � k � m1 and that there exist scalars λij ∈ F,1 � i < j � m1, such that either

φ(E

(m1)ij ⊗X

)= λijE

(m1)ij ⊗ π(X), X ∈ Mm2···ml

,

or


φ(E

(m1)ij ⊗X

)= λijE

(m1)ji ⊗ π(X), X ∈ Mm2···ml

,

where π = π1.

Proof. Without loss of generality, we may assume that i = 1 and j = 2 (the proof forother cases goes through in the same way). Set F1 := E

(m1)11 +E

(m1)12 , F2 := E

(m1)22 −E

(m1)12 ,

and Fk := E(m1)kk for k = 3, . . . ,m1. Then, by Claim 3, there exists an invertible Q ∈

Mm1···mlsuch that for all X ∈ Mm2···ml

, we have

φ(Fk ⊗X) = Q(E

(m1)kk ⊗ πk(X)

)Q−1, k = 1, . . . ,m1, (5)

where πk are canonical maps on Mm2···ml. This, together with (4), implies

E(m1)kk ⊗ πk(X) = Q

(E

(m1)kk ⊗ πk(X)

)Q−1, k = 3, . . . ,m1. (6)

Let Q = [Qij ]m1×m1 with Qij ∈ Mm2···mland let X = Im2···ml

. It follows from (6) that

(E


)Q = Q

(E


), k = 3, . . . ,m1.

Hence,

Q =[Q11 Q12Q21 Q22

]⊕ Diag(Q33, . . . , Qm1m1). (7)

Suppose that

[Q11 Q12Q21 Q22

]−1

=[T11 T12T21 T22

]

for some T11, T12, T21, T22 ∈ Mm2···ml. It follows from (4), (5), and (7) that

φ(E

(m1)12 ⊗X

)=

[Q11π1(X)T11 − π1(X) Q11π1(X)T12

Q21π1(X)T11 Q21π1(X)T12

]⊕ 0r (8)

for all X ∈ Mm2···mland with r = (m1 − 2)m2 · · ·ml. Since for any idempotent P =

P2⊗· · ·⊗Pl ∈ Pm2···mland any scalar λ ∈ F, the matrix (E(m1)

11 +λE(m1)12 )⊗P ∈ Mm1···ml

is an idempotent, we obtain, using (4) and (8), that[π1(P ) 0

0 0

]+ λ

[Q11π1(P )T11 − π1(P ) Q11π1(P )T12

Q21π1(P )T11 Q21π1(P )T12

]∈ P2m2···ml

.

Since λ was an arbitrary scalar, it follows that Q21π1(P )T12 = 0 and, since P was anarbitrary idempotent which can be written as a tensor product of matrices, one canconclude that Q21XT12 = 0 for all X ∈ Mm2···ml

(here we used the form of π1). Hence,either Q21 = 0 or T12 = 0.


Case I. First, suppose that Q21 = 0. Using the identity[Q11 Q120 Q22

]−1

=[Q−1

11 −Q−111 Q12Q

−122

0 Q−122

]

and (8), it follows that

φ(E

(m1)12 ⊗X

)=

[Q11π1(X)Q−1

11 − π1(X) −Q11π1(X)Q−111 Q12Q

−122

0 0

]⊕ 0r (9)

for all X ∈ Mm2···ml. Note that if P = P2 ⊗ · · · ⊗ Pl ∈ Pm2···ml

and λ ∈ F, then(E(m1)

22 + λE(m1)12 )⊗P ∈ Mm1···ml

is an idempotent as well. Thus, using (4) and (9), weconclude that[

0 00 π2(P )

]+ λ

[Q11π1(P )Q−1

11 − π1(P ) −Q11π1(P )Q−111 Q12Q

−122

0 0

]

is an idempotent from M2m2···ml. Since the above observation holds true for any scalar λ,

it follows that Q11π1(P )Q−111 = π1(P ). Furthermore, by Lemma 2.3, we have π1 = π1

and T11 = εIm2···mlfor some scalar ε = 0. Set U := −Q12Q

−122 . Then

φ(E

(m1)12 ⊗X

)= E

(m1)12 ⊗ π1(X)U, X ∈ Mm2···ml

. (10)

Next, if P = P2 ⊗ · · · ⊗ Pl ∈ Pm2···ml, then (E(m1)

12 +E(m1)22 )⊗ P ∈ Mm1···ml

is again anidempotent and, thus, using (4) and (10), we conclude that

[0 π1(P )U0 π2(P )

]∈ P2m2···ml

.

Hence, π1(P )U = π1(P )Uπ2(P ) for all idempotents in Mm2···mlwhich can be written

as a tensor product of matrices. Now, Lemma 2.4 implies that U = λ12Im2···mlfor some

scalar λ12. In Step 2 we show that λ12 = 0. So, assume for a moment that we have alreadyproved this. Then, again using Lemma 2.4, one can see that π1 = π2 = π. Therefore,

φ(E

(m1)12 ⊗X

)= λ12E

(m1)12 ⊗ π(X), X ∈ Mm2···ml

.

Case II. If T12 = 0, then, using a similar approach as in Case I, we show that there existsa scalar λ12 ∈ F such that

φ(E

(m1)12 ⊗X

)= λ12E

(m1)21 ⊗ π(X), X ∈ Mm2···ml

,

where π = π1 = π2. �By Step 1 and (4), we may assume that

φ(E

(m1)ij ⊗X

)= λijτij

(E

(m1)ij

)⊗ π(X) (11)


for all X ∈ Mm2···mland 1 � i, j � l, where π is a canonical map on Mm2···ml

and for1 � i, j � m1, τij is the identity map X �→ X or the transposition map X �→ Xt.

Step 2. We show that for all 1 � i < j � m1, we have

τij = τji and λijλji = 1.

Proof. Note that(

12E

(m1)ii + 1

2E(m1)ij + 1

2E(m1)ji + 1

2E(m1)jj

)⊗ Im2···ml

is an idempotent for all 1 � i < j � m1. Thus, according to (11), it follows that thematrix

(12E

(m1)ii + 1

2λijτij(E

(m1)ij

)+ 1

2λjiτji(E

(m1)ji

)+ 1

2E(m1)jj

)⊗ Im2···ml

is an idempotent as well. Therefore, τij = τji and λijλji = 1 for all 1 � i < j � m1, asdesired. �Step 3. If m1 � 3, then τij = τ12 and λijλjk = λik for distinct 1 � i, j, k � m1.

Proof. Note that (E(m1)ii + E

(m1)ij + E

(m1)ik ) ⊗ Im2···ml

is an idempotent for distinct 1 �i, j, k � m1. Thus, using (11), it follows that the matrix

(E

(m1)ii + λijτij

(E

(m1)ij

)+ λikτik

(E

(m1)ik

))⊗ Im2···ml

is an idempotent. This implies that τij = τik. Furthermore, since (E(m1)ij + E

(m1)ik +

E(m1)jj + E

(m1)jk ) ⊗ Im2···ml

is an idempotent, we conclude, using (11), that the matrix

(λijτij

(E

(m1)ij

)+ λikτik

(E

(m1)ik

)+ E

(m1)jj + λjkτjk

(E

(m1)jk

))⊗ Im2···ml

is an idempotent as well. This, together with τij = τik, implies that τij = τjk andλijλjk = λik for all distinct 1 � i, j, k � m1, as desired. �

Denote τ1 = τ12 and set

Q = Diag(1, λ−1

12 , . . . , λ−11m1

)⊗ Im2···ml

.

Then

φ(A1 ⊗X) = Q(τ1(A1) ⊗ π(X)

)Q−1

for all A1 ∈ Mm1 and all X ∈ Mm2···ml. This completes the proof of the Main Theorem.


Acknowledgements

Jinli Xu was supported by the National Natural Science Foundation of China (GrantNo. 11171294) and the Natural Science Foundation of Heilongjiang Province of China(Grant No. A201013). This research was done when Ajda Fošner and Jinli Xu wereattending the Summer Research Workshop on Quantum Information Science 2013 atTaiyuan University of Technology. They gratefully acknowledge the support and kindhospitality from the host university.

References

[1] N. Bourbaki, Algebra I, Elements of Mathematics, Springer-Verlag, New York, 1989.[2] M. Brešar, M.A. Chebotar, W.S. Martindale III, Functional Identities, Birkhäuser, Basel, 2007.[3] G.-H. Chan, M.-H. Lim, Linear preservers on powers of matrices, Linear Algebra Appl. 162–164

(1992) 615–626.[4] J. Dieudonné, Sur une généralisation du groupe orthogonal à quatre variables, Arch. Math. 1 (1949)

282–287.[5] A. Fošner, Z. Huang, C.-K. Li, Y.-T. Poon, N.-S. Sze, Linear maps preserving the higher numerical

ranges of tensor products of matrices, Linear Multilinear Algebra (2013), http://dx.doi.org/10.1080/03081087.2013.790386, 16 pp.

[6] A. Fošner, Z. Huang, C.-K. Li, N.-S. Sze, Linear preservers and quantum information science, LinearMultilinear Algebra 61 (10) (2013) 1377–1390.

[7] A. Fošner, Z. Huang, C.-K. Li, N.-S. Sze, Linear maps preserving numerical radius of tensor productsof matrices, J. Math. Anal. Appl. 407 (2013) 183–189.

[8] A. Fošner, Z. Huang, C.-K. Li, N.-S. Sze, Linear maps preserving Ky Fan norms and Schatten normsof tensor products of matrices, SIAM J. Matrix Anal. Appl. 34 (2013) 673–685.

[9] S. Friedland, C.-K. Li, Y.-T. Poon, N.-S. Sze, The automorphism group of separable states inquantum information theory, J. Math. Phys. 52 (2011), http://dx.doi.org/10.1063/1.3578015, 8 pp.

[10] G. Frobenius, Über die Darstellung der endlichen Gruppen durch Linear Substitutionen, Sitzungs-ber. Dtsch. Akad. Wiss. Berl. (1897) 994–1015.

[11] N. Jacobson, Lectures in Abstract Algebra, vol. II, Van Nostrand, New York, 1953.[12] N. Johnston, Characterizing operations preserving separability measures via linear preserver prob-

lems, Linear Multilinear Algebra 59 (2011) 1171–1187.[13] A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, New Jersey, 1989.[14] S.M. Lane, B. Birkhoff, Algebra, AMS Chelsea, Providence, 1999.[15] C.-K. Li, S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001) 591–605.[16] L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function

Spaces, Lecture Notes in Mathematics, vol. 1895, Springer, Berlin, 2007.[17] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge Univer-

sity Press, Cambridge, 2000.[18] L. Shaowu, Linear maps preserving idempotence on matrix modules over principal ideal domains,

Linear Algebra Appl. 258 (1997) 219–231.[19] W.-H. Steeb, Y. Hardy, Matrix Calculus and Kronecker Product: A Practical Approach to Linear

and Multilinear Algebra, 2nd edition, World Scientific, Singapore, 2011.[20] Z.-X. Wan, Geometry of Matrices, World Scientific Publishing Co., New York, 1996.

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib42s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib427265736172436865626F7461724D617274696E64616C65s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib434C31393932s1


http://refhub.elsevier.com/S0024-3795(14)00066-4/bib44696575646F6E6E65s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib44696575646F6E6E65s1

http://dx.doi.org/10.1080/03081087.2013.790386







http://dx.doi.org/10.1063/1.3578015

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib46726F62656E696F7573s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib46726F62656E696F7573s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4A31393533s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4As1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4As1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4A61696Es1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4Ds1


http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4D6F6C6E6172s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4D6F6C6E6172s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4E43s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib4E43s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib5368616F7775s1

http://refhub.elsevier.com/S0024-3795(14)00066-4/bib5368616F7775s1



http://refhub.elsevier.com/S0024-3795(14)00066-4/bib57616Es1

http://dx.doi.org/10.1080/03081087.2013.790386

Linear maps preserving idempotents of tensor products of matrices

Documents

Transcript of Linear maps preserving idempotents of tensor products of matrices