Given an endomorphism A over a finite dimensional vector space having
Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant
subspaces of A can be obtained from the nilpotent part of this decomposition.
We extend this result for lattices of characteristic subspaces. We also obtain
a generalization of Shoda's Theorem about the characterization of the existence
of characteristic non hyperinvariant subspaces

Mingueza, David

Montoro, M. Eulàlia

Roca, Alicia

English

arXiv

[EN] Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's theorem about the characterization of the existence of characteristic non hyperinvariant subspaces. (C) 2018 Elsevier Inc. All rights reserved.The second author is partially supported by grant MTM2015-65361-P MINECO/FEDER, UE and MTM2017-90682-REDT. The third author is partially supported by grants MTM2017-83624-P and MTM2017-90682-REDT.Mingueza, D.; Montoro, ME.; Roca Martinez, A. (2018). The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition. Linear Algebra and its Applications. 558:63-73. https://doi.org/10.1016/j.laa.2018.08.005637355

Roca Martinez, Alicia

RiuNet

 Document downloaded from:  This paper must be cited as:                  The final publication is available at   Copyright  Additional Information  http://hdl.handle.net/10251/125132Mingueza, D.; Montoro, ME.; Roca Martinez, A. (2018). The lattice of characteristicsubspaces of an endomorphism with Jordan-Chevalley decomposition. Linear Algebra andits Applications. 558:63-73. https://doi.org/10.1016/j.laa.2018.08.005http://doi.org/10.1016/j.laa.2018.08.005ElsevierThe lattice of characteristic subspaces of anendomorphism with Jordan-ChevalleydecompositionDavid MinguezaAccenture, 08022 Barcelona, Spain ∗M. Eulàlia MontoroDepartamento de Matemáticas e Informática. Universitat de Barcelona †Alicia RocaDepartamento de Matemática Aplicada, IMM,Universitat Politècnica de València ‡AbstractGiven an endomorphism A over a finite dimensional vector space hav-ing Jordan-Chevalley decomposition, the lattices of invariant and hyper-invariant subspaces of A can be obtained from the nilpotent part of thisdecomposition. We extend this result for lattices of characteristic sub-spaces. We also obtain a generalization of Shoda’s Theorem about thecharacterization of the existence of characteristic non hyperinvariant sub-spaces.Keywords: Hyperinvariant subspaces, characteristic subspaces, lattices.1 IntroductionThe lattice of characteristic subspaces of an endomorphism over a finite di-mensional space has been studied in [1, 2, 7, 8], where structural properties ofthe lattice have been given when the minimal polynomial of the endomorphismsplits over the underlying field F. It was proved in ([1]) that only if F = GF (2),the lattices of characteristic and hyperinvariant subspaces may not coincide.When the minimal polynomial of the endomorphism does not split over F, thelattice of characteristic subspaces has not been described. The aim of this paper∗david.mingueza@outlook.es†eula.montoro@ub.edu‡aroca@mat.upv.es1is to analyze this case when the minimal polynomial of the endomorphism isseparable.The results are based on the Jordan-Chevalley decomposition of an endomor-phism which exists if the minimal polynomial of the endomorphism is separable(see [5, 9]). In particular, on perfect fields every irreducible polynomial is sepa-rable, therefore the Jordan-Chevalley decomposition holds.The paper is organized as follows: in Section 2 we introduce some definitionsand previous results. We present some lemmas showing that the study thelattices of the invariant and hyperinvariant subspaces of an endomorphism canbe reduced to the case of an endomorphism where the minimal polynomial is apower of an irreducible polynomial p.In Section 3, out of the Jordan-Chevalley decomposition of an endomorphisminto its commuting semisimple and nilpotent parts, we reduce the problem tothe study of the lattice of characteristic subspaces of the associated nilpotentpart. Finally, we conclude that characteristic non hyperinvariant subspacescan only appear when the irreducible factor of the minimal polynomial is ofdegree 1, in other words, if p splits over F. This result extends Shoda’s theorem(Theorema 2.9) for the separable case.2 PreliminariesWe introduce some definitions and previous results which will be used through-out the paper.Let F be a field. Let V be an n-dimensional vector space over F and f :V → V an endomorphism. We denote by A ∈Mn(F) its associated matrix withrespect to given basis and by mA the minimal polynomial of A. In what followswe will identify f with A. The degree of a polynomial p is written as deg(p),and | · | is the “cardinality of”.The lattice of the vector subspaces of V will be denoted by  LF(V ) .A subspace Y ⊆ V is invariant with respect to A if AY ⊆ Y . We denote byInvF(A) the lattice of invariant subspaces.The centralizer of A over F is the algebra ZF(A) = {B ∈ Mn(F) : AB =BA}. If we only take those endomorphisms B that are automorphisms we writeZ∗F(A) = {B ∈Mn(F) : AB = BA, det(B) 6= 0}.A subspace Y ⊆ V is called hyperinvariant with respect to A if BY ⊆ Y forall matrices B ∈ ZF(A). We denote by HinvF(A) the lattice of hyperinvariantsubspaces.An invariant subspace Y ⊆ V with respect to A is called characteristic ifBY ⊆ Y for all matrices B ∈ Z∗F(A) and AY ⊆ Y . We denote by ChinvF(A)the lattice of characteristic subspaces.Obviously,HinvF(A) ⊆ ChinvF(A) ⊆ InvF(A).2The lattices of invariant and hyperinvariant subspaces allow the followingdecompositions.Proposition 2.1. [3] Let A and B be endomorphisms on finite dimensionalvector spaces V and W respectively, over a field F. The following properties areequivalent:1. The minimum polynomials of A and B are relatively prime.2. InvF(A⊕B) = InvF(A)⊕ InvF(B).Proposition 2.2. [4] Let A and B be endomorphisms on finite dimensionalvector spaces V and W respectively, over a field F. The following properties areequivalent:1. The minimum polynomials of A and B are relatively prime.2. HinvF(A⊕B) = HinvF(A)⊕HinvF(B).As a consequence, if mA = pk11 · · · pkrr is the prime decomposition of mA fora given A ∈Mn(F), taking into account the primary decompositionV = V1 ⊕ · · · ⊕ Vr, Vi = ker(pi(A)ki), i = 1, . . . , r,and Ai = A|Vi, i = 1, . . . r, the next result is satisfied (see [3, 4]).Proposition 2.3. Let A ∈Mn(F) and mA = pk11 · · · pkrr with p1, . . . , pr distinctirreducible polynomials. Then,1. InvF(A) = InvF(A1)⊕ · · · ⊕ InvF(Ar).2. HinvF(A) = HinvF(A1)⊕ · · · ⊕HinvF(Ar).If p(x) ∈ F[x] is an irreducible polynomial and α /∈ F is a root of p(x), wedenote by F(α) the minimal extension of F containing α. The next result canbe found in [3]. Although a proof of the following property 4 was included in[3], we give here a simple proof of the whole lemma.Lemma 2.4. [3] Let A ∈Mn(F). Assume that mA = p0+p1x+. . .+psxs ∈ F[x]is irreducible, and that α is a root of p such that α /∈ F. Let K = {a0In + a1A+. . .+as−1As−1 |ai ∈ F} be the algebra of polynomials of degree at most s. Then,1. K is a field isomorphic to F(α).2. V is a K−vector space.3. A is K−linear.4. InvF(A) =  LK(V ).3Proof. 1. The following applicationΓ : K −→ F(α),with Γ(a0In + a1A . . . + as−1As−1) = a0 + a1α + . . . + as−1αs−1 is anisomorphism.2. For all λ = a0In + a1A+ . . .+ as−1As−1 ∈ K, it is satisfied thatλv = (a0In + a1A+ . . .+ as−1As−1)v ∈ V, ∀v ∈ V.3. For all λ ∈ K, Aλv = λAv.4. The following equivalences hold,W ∈ InvF(A)⇔ AW ⊆W ⇔ λW ⊆W, ∀λ ∈ K⇔W ∈  LK(V ).Notice that properties 1− 4 of this lemma are trivially true if s = 1.A polynomial is called p-primary if it is of the form pr for some irreduciblepolynomial p and a positive integer r. A polynomial p(x) over a field F is calledseparable if its roots are distinct in an algebraic closure of F.A linear operator S on a finite-dimensional vector space is semisimple ifevery S-invariant subspace has a complementary S-invariant subspace. A linearoperator N on a finite-dimensional vector space is nilpotent if Nk = 0 for somepositive integer k.The next lemma contains the Jordan-Chevalley decomposition of a matrix,which plays a key role in this work. The result can be found in [9].Lemma 2.5. (Jordan-Chevalley decomposition) Let A ∈ Mn(F) be a matrixwith mA = pr, where p is irreducible and separable. Then, there is a uniquedecompositionA = S +N,where S is semisimple, N is nilpotent and SN = NS. Moreover, S and N arepolynomials in A.Remark 2.6. [3]If A has Jordan-Chevalley decomposition then there exist a transformationinto a rational canonical form such that A is similar todiag(C1, C2, . . . , Cm),withCi =C 0 . . . 0 0Is C . . . 0 0....... . .......0 0 . . . C 00 0 . . . Is C ∈Mni(F), i = 1, 2, . . . ,m,4where C is the companion matrix of p, and n1 ≥ n2 ≥ · · · ≥ nm > 0, n1 + . . .+nm = n, n1 = sr (where s = deg(p)), thenS = diag(S1, S2, . . . , Sm), N = diag(N1, N2, . . . , Nm),whereSi =C 0 . . . 0 00 C . . . 0 0....... . .......0 0 . . . C 00 0 . . . 0 C , Ni =0 0 . . . 0 0Is 0 . . . 0 0....... . .......0 0 . . . 0 00 0 . . . Is 0 .Taking advantage of the Jordan-Chevalley decomposition it can be provedthat the lattices of the invariant and hyperinvariant subspaces of A over F canbe obtained as lattices of the corresponding subspaces of a nilpotent matrixover a different field K (see [3] and [4], respectively). The results are includedin the next theorem. Although proofs are sketched in [3, 4], we provided themin detail.Theorem 2.7. Let A ∈ Mn(F). Assume that mA is p−primary (mA = pr)with p separable and deg(p) = s. Let A = S + N be the Jordan-Chevalleydecomposition of A. Let K = {a0In + a1S + . . .+ as−1Ss−1 |ai ∈ F}. Then,1. InvF(A) = InvK(A) = InvK(N).2. HinvF(A) = HinvK(A) = HinvK(N).Proof. 1. W ∈ InvF(A)⇔W ∈ InvK(A) (becauseA is K− linear). Moreover,since S ∈ KInvK(A) = InvK(A− S) = InvK(N).2. Notice thatZK(A) = ZF(A) ∩ ZF(S) = ZF(A),andZK(A) = ZK(A− S) = ZK(N).ThereforeHinvF(A) = HinvK(A) = HinvK(A− S) = HinvK(N).Next example illustrates the above lemma for the field of real numbers.Example 2.8. Let F = R, V = R4 andA =0 1 0 0−1 0 0 01 0 0 10 1 −1 0 .5The minimal polynomial of A is mA = (x2 + 1)2. The Jordan-Chevalley decom-position of A is given byS =0 1 0 0−1 0 0 00 0 0 10 0 −1 0 , N =0 0 0 00 0 0 01 0 0 00 1 0 0 ,and K = {a0I4 + a1S, ai ∈ R}. Notice that the Jordan chains of N aree1 → e3 → 0,e2 → e4 → 0.Then,InvR(A) = InvK(N) = {0, 〈e3, e4〉, V } .The lattice of hyperinvariant subspaces isHinvR(A) = HinvK(N) = {0, 〈e3, e4〉, V }.Concerning to characteristic subspaces, Shoda’s theorem characterizes theexistence of characteristic non hyperinvariant subspaces.Theorem 2.9. [10] Let V be a finite-dimensional vector space over the fieldF = GF (2) and let f : V → V be a nilpotent linear operator. The followingstatements are equivalent:1. There exists a characteristic subspace of V which is not hyperinvariant.2. For some numbers r and s with s > r + 1 the Jordan form of f containsexactly one Jordan block of size s and exactly one block of size r.Astuti-Wimmer proved ([1]) that characteristic non hyperinvariant subspacescan only exist on the field GF (2).Theorem 2.10. [1] Let V be a finite dimensional vector space over a field Fand let f : V → V be a linear operator. Assume that the minimal polynomial off splits over F. If |F| > 2, then ChinvF(f) = HinvF(f).3 Reduction to the nilpotent case for character-istic subspacesIn this section we focus on the study of the lattice of characteristic subspacesof an endomorphism over a field when the irreducible factors of the minimalpolynomial are separable.First of all we see that the general case can be reduced to the especific case ofendomorphisms having minimal polynomials with an unique irreducible factor,as in the lattices of invariant of hyperinvariant subspaces.6Lemma 3.1. Let A and B be endomorphisms on finite dimensional vectorspaces V and W respectively, over a field F. The following properties are equiv-alent:1. The minimum polynomials of A and B are relatively prime.2. ChinvF(A⊕B) = ChinvF(A)⊕ ChinvF(B).Proof. First assume that (1) is true. It is evident that Chinv(A⊕B) ⊆ Chinv(A)⊕Chinv(B) holds even for an arbitrary polynomials.As it is stated in [11], Z(A⊕B) = Z(A)⊕Z(B), therefore every commutingautomorphism of A⊕B is of the form X =(X1 00 X2)with X1 ∈ Z∗(A) andX2 ∈ Z∗(B).Taking W = W1 ⊕W2 ∈ Chinv(A)⊕ Chinv(B), clearly W ∈ Chinv(A⊕B)follows.Assume now that (2) is true.As Chinv(A ⊕ B) = Chinv(A) ⊕ Chinv(B), it is satisfied that V ⊕ 0 ∈Chinv(A ⊕ B). Notice that if X is a matrix such that XA = BX, then(I 0X I)∈ Z∗(A ⊕ B). Therefore, V ⊕ 0 must be invariant for this matrixand this implies X = 0.If gcd(mA,mB) = q, deg(q) > 0, then mA = qf and mB = qh for somepolynomials f, h, and w.l.o.g. we can suppose gcd(q, f) = 1. With respect to anappropriate basis, we can write A =(Cq 00 F)and B =(Cq H0 G)whereCq is the companion matrix of q and for some matrices F,H and G.Let U ∈ Z∗(Cq), then X =(U 00 0)6= 0 satisfies XA = BX, which isclearly a contradiction.Corollary 3.2. Let A ∈Mn(F) and mA = pk11 . . . pkrr with pi distinct irreduciblepolynomials. Then,Chinv(A) = Chinv(A1)⊕ · · · ⊕ Chinv(Ar), (1)where Ai ∈ Mni(F) is the restriction of A ∈ Mn(F) to Vi = ker((pi(A))ki),i = 1, 2, . . . , r, (V = V1 ⊕ · · · ⊕ Vr).In the next lemma we see that the lattice of characteristic subspaces of Acan also be determined on a different field for a nilpotent matrix, using Jordan-Chevalley decomposition.Lemma 3.3. Let A ∈Mn(F). Assume that mA is p−primary (mA = pr) with pseparable and deg(p) = s. Let A = S+N be the Jordan-Chevalley decompositionof A. Let K = {a0In + a1S + . . .+ as−1Ss−1 |ai ∈ F}. Then,1. ChinvF(A) = ChinvK(A) = ChinvK(N).7Proof. Similarly to the proof of Theorem 2.7,Z∗K(A) = Z∗F(A) ∩ Z∗F(S) = Z∗F(A),andZ∗K(A) = Z∗K(A− S) = Z∗K(N).ThereforeChinvF(A) = ChinvK(A) = ChinvK(A− S) = ChinvK(N).Finally we obtain the conclusion that characteristic non hyperinvariant sub-spaces can only exist over Jordan blocks.Theorem 3.4. Let A be an endomorphism on a finite dimensional space Vover a field F. Assume that mA is p−primary (mA = pr) with p separable. IfChinvF(A) \HinvF(A) 6= ∅, then deg(p) = 1 and F = GF (2).Proof. Let us assume that ChinvF(A) \ HinvF(A) 6= ∅ and deg(p) > 1. As pis separable, let A = N + S be the Jordan-Chevalley decomposition of A. ByLemma 3.3 we know that ChinvF(A) = ChinvK(N) with K defined as in Lemma3.3. By assumption deg(p) > 1, which implies that |K| > 2, and by [1, Theorem3.4], ChinvK(N) = HinvK(N). Finally, by Theorem 2.7, HinvK(N) = HinvF(A).Therefore, ChinvF(A) = HinvF(A), which is a contradiction.We see that for the existence of characteristic non-hyperinvariant subspacesit is necessary that deg(p) = 1 and F = GF (2). If we take into account theShoda condition, we obtain a necessary and sufficient condition.Corollary 3.5. Let A be an endomorphism on a finite dimensional space Vover a field F. Assume that mA is p−primary (mA = pr) with p separable.Then, the next two statement are equivalent:1. ChinvF(A)\HinvF(A) 6= ∅.2. F = GF (2), deg(p) = 1 and the Shoda condition is satisfied.We show next how the lattice of characteristic subspaces can be obtained intwo examples. In the first one HinvF(A) = ChinvF(A), and in the second oneChinvF(A) 6⊆ HinvF(A) for some matrices A.Example 3.6. Let F = GF (2) andA =0 1 0 0 0 0 0 01 1 0 0 0 0 0 01 0 0 1 0 0 0 00 1 1 1 0 0 0 00 0 1 0 0 1 0 00 0 0 1 1 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 1.8The minimal polynomial of A is p-primary, mA = (x2 + x+ 1)3, p = x2 + x+ 1separable, and the Jordan-Chevalley decomposition of A isS =0 1 0 0 0 0 0 01 1 0 0 0 0 0 00 0 0 1 0 0 0 00 0 1 1 0 0 0 00 0 0 0 0 1 0 00 0 0 0 1 1 0 00 0 0 0 0 0 0 10 0 0 0 0 0 1 1, N =0 0 0 0 0 0 0 00 0 0 0 0 0 0 01 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0.Let K = {a0I8 + a1S, ai ∈ GF (2)}.The Segre characteristic of N is (3, 3, 1, 1) with Jordan chainse1 → e3 → e5 → 0e2 → e4 → e6 → 0e7 → 0e8 → 0The characteristic and hyperinvariant subspaces of N are (see [7]):HinvK(N) = ChinvK(N) = {0, 〈e3, e5, e4, e6, e7, e8〉, 〈e3, e5, e4, e6〉, 〈e5, e6, e7, e8〉, 〈e5, e6〉, V } .Example 3.7. Let F = GF (2) andA =1 0 0 01 1 0 00 1 1 00 0 0 1 .The minimal polynomial of A is mA = (x + 1)3, p = x + 1 trivially splits overF andS =1 0 0 00 1 0 00 0 1 00 0 0 1 , N =0 0 0 01 0 0 00 1 0 00 0 0 0Observe that K = F = GF (2) as deg(p) = 1.The Segre characteristic of N is (3, 1) with vector chains:e1 → e2 → e3 → 0e4 → 0The hyperinvariant and characteristic non hyperinvariant subspaces, accordingto [7], are:HinvK(N) = {0, 〈e2, e3, e4〉, 〈e2, e3〉, 〈e3, e4〉, 〈e3〉, V } ,ChinvK(N)\HinvK(N) = {e2 + e4 + 〈e3〉} .9AcknowledgmentsThe second author is partially supported by grant MTM2015-65361-P MINECO/FEDER,UE and MTM2017-90682-REDT. The third author is partially supported bygrants MTM2017-83624-P and MTM2017-90682-REDT.References[1] P. Astuti, H.K. Wimmer. Hyperinvariant, characteristic and marked sub-spaces. Oper. Matrices 3, (2009) 261-270.[2] P. Astuti, H.K. Wimmer. Characteristic and hyperinvariant subspaces overthe field GF(2). Linear Algebra Appl. 438, 4, (2013), 1551-1563.[3] L. Brickman, P.A. Fillmore. The invariant subspace lattice of a linear trans-formation. Can. J. Math., 19, 35, (1967), 810-822.[4] P.A. Fillmore, D.A. Herrero and W.E. Longstaff. The hyperinvariant sub-space lattice of a linear transformation. Linear Algebra Appl. 17 (1977),125-132.[5] K.Hoffman and R. Kunze. Linear Algebra. Second Edition, Prentice-Hall,Englewood Cliffs, N.J.,1971.[6] 2002]Lang02S. Lang. Algebra, Vol. I, Princeton, N.J., 1953.[7] D. Mingueza, M.E. Montoro, J.R. Pacha. Description of the characteristicnon-hyperinvariant subspaces over the field GF (2). Linear Algebra Appl.439 (2013), 3734-3745.[8] D. Mingueza, M.E. Montoro, A. Roca. The characteristic subspace latticeof a linear transformation. Linear Algebra Appl. 506 (2016) 329-341.[9] C. Norman. Finitely Generated Abelian Groups and Similarity of Matricesover a Field. Springer-Verlag, London Ltd. 2012.[10] K. Shoda. Über die characteristischen Untergruppen einer endlichenAbelschen Gruppe. Math. Z. 31 (1930) 611–624.[11] D. A. Suprunenko,R. I. Tyshkevich. Commutative Matrices. Academic Pa-perbacks, 1968.10

The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition

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