Journal of Mathematical Sciences, Vol. 186, No. 3, October, 2012
NONLINEAR MAPPINGS WHOSE JACOBI MATRIX COMMUTES WITH CONSTANT MATRICES OF A RING Yu. A. Chirkunov Novosibirsk State Technical University 20, pr. Karla Marksa, Novosibirsk 630092, Russia
[email protected]
UDC 517.944 + 519.46
We obtain a criterion for the existence of a nonlinear mapping u : C m → C m (m 2) whose Jacobi matrix commutes with each constant complex matrix of a given ring Q. We show that such a mapping exists if and only if the ring Q has at least one (r, l)-pair. Bibliography: 4 titles.
Introduction We obtain a criterion of the existence of a nonlinear mapping u : C m → C m (m 2) such that its Jacobi matrix commutes with every matrix of an arbitrarily given ring Q of constant quadratic matrtices of order m over the field of complex numbers. The main result asserts that such a mapping exists if and only if all matrices in Q have the common left and right eigenvectors corresponding to the same eigenvalue for each matrix of this ring. Such a problem arises, for example, in the group analysis of differential equations, when we study the dependence of transformations of the basic Lie group on independent variables [1]. The main algebraic notions and assertions can be found in [2]–[4].
1
Statement of the Problem
Let Q = {A} be a ring of constant quadratic matrices of order m (m 2) over the field of complex numbers. The question arises whether there exists a nonlinear mapping u : C m → C m such that its Jacobi matrix ux = ux (x), x ∈ C m , commutes with every matrix of the ring Q, i.e., satisfies the system of equations Aux = ux A
for all matrices A ∈ Q.
(1)
The equivalence of transformations of this problem is determined by the following assertion which can be proved by an immediate calculation. Lemma 1. The system of equations (1) is invariant under the transformations u = Bu ,
x = Bx ,
(2)
Translated from Vestnik Novosibirskogo Gosudarstvennogo Universiteta: Seriya Matematika, Mekhanika, Informatika 10, No. 1, 2010, pp. 108-118. c 2012 Springer Science+Business Media, Inc. 1072-3374/12/1863-0379
379
where B is a nonsingular matrix. The matrix of the system (1) is transformed by the formula A = B −1 AB. Definition 1. We say that a solution u = u(x) to the system (1) is nontrivial if uxx = 0. The existence of a nontrivial solution to the system (1) is equivalent to the existence of a matrix K = K(x) satisfying the following three conditions: 1) commutability AK = KA for all A ∈ Q, (3) 2) consistency ∂τ y σ = ∂σ y τ ,
σ, τ = 1, 2, . . . , m,
(4)
where yσ = (yσ1 , yσ2 , . . . , yσm )T is the σth column of the matrix K, ∂τ =
∂ (τ = 1, 2, . . . , m) , ∂xτ
T x = x1 , x2 , . . . , xm ,
3) nontriviality ∂K = 0,
∂ = (∂1 , ∂2 , . . . , ∂m ).
(5)
A direct computation yields the following assertion. Lemma 2. The conditions (3)–(5) are invariant under the choice of the basis for C m . By this lemma, to find a matrix K = K(x) satisfying (3)–(5), instead of the ring Q = {A} −1 we can consider any equivalent ring B AB . Definition 2. A column-vector r=0 and a row-vector l=0 are referred to as an (r, l) pair of a ring Q of quadratic matrices of order m (the ring Q possesses an (r, l)-pair) if for every matrix A ∈ Q we have Ar = λ(A)r and lA = λ(A)l, where λ(A) ∈ C is an eigenvalue of the matrix A. If matrices {A} possess an (r, l)-pair, then the matrices {A } obtained by an equivalence transformation of the form (2) possess a (B −1 r, lB)-pair. Proposition 1. A ring of matrices possesses at least one (r, l)-pair if and only if there exists a matrix of rank 1 that commutes with every matrix of this ring.
2
The Main Result
Theorem 1 (the main theorem). A matrix K = K(x) satisfying (3)–(5) exists if and only if the ring Q possesses at least one (r, l)-pair. Proof. Sufficiency. Assume that the ring Q possesses an (r, l)-pair. Then the matrix M = r ⊗ l commutes with all matrices of the ring Q. For K = K(x) we take the matrix K(x) = h(x)M . From (4) it follows that h(x) = H(l · x), where H is an arbitrary scalar function. The nontriviality condition (5) is satisfied if H = 0. Necessity follows from Theorem 2 below. Theorem 2. If a ring Q possesses no (r, l)-pairs, then a matrix K = K(x) satisfying (3) and (4) is constant. 380
Proof. 1. Irreducible ring. If a ring Q is irreducible, then K(x) = h(x)E by the Schur lemma [4], where h(x) is a scalar function . By (4), δσα ∂τ h = δτα ∂σ h (α, σ, τ = 1, 2, . . . , m). Therefore, ∂τ h = 0 for all τ = 1, 2, . . . , m. Consequently, K = K(x) is a constant matrix. 2. Reducible ring. Assume that Q is a reducible ring and Pi = dim Vi , ni = dim(Vi /Vi−1 ) = Pi − Pi−1 (i = 1, 2, . . . , p), where (0) = V0 ⊂ V1 ⊂ . . . ⊂ Vp = C m
(6)
is the composition series of the ring Q. If (l1 , l2 , . . . , lm ) is a basis for the space C m such that (l1 , l2 , . . . , lpi ) is a basis for the invariant subspace Vi (i = 1, 2, . . . , p). Then the matrices of a ring equivalent to Q take in this basis the quasitriangular form ⎞ ⎛ 1 A1 A12 . . . A1p ⎜ 0 A2 . . . A2 ⎟ ⎜ p ⎟ 2 A=⎜ .. .. ⎟ ⎟, ⎜ .. . ... . ⎠ ⎝ . 0 0 . . . App where the matrix Aij consists of ni rows and nj columns; moreover, Aij = 0 for i > j. The set Qi = {Aii : A ∈ Q} is irreducible for all i = 1, 2, . . . , p. The subdivision of the set of indices [1, m] =
p
Ji ,
Ji = (Pi−1 , Pi ]
i=1
leads to the division of the matrix K into blocks ⎛ 1 K1 K21 ⎜K 2 K 2 ⎜ 1 2 K=⎜ .. ⎜ .. . ⎝ . p K1 K2p
⎞ . . . Kp1 . . . Kp2 ⎟ ⎟ .. ⎟ ⎟, ... . ⎠ . . . Kpp
where the matrix Kβα consists of yji (i ∈ Jα , j ∈ Jβ ). The condition (3) takes the form Aασ Kβσ = Kσα Aσβ
(α, β = 1, 2, . . . , p),
(7)
T (β = 1, 2, . . . , p) and which can be written in terms of the columns Kβ = Kβ1 , Kβ2 , . . . , Kβp α α α α rows K = (K1 , K2 , . . . , Kp ) (α = 1, 2, . . . , p) as follows: AKβ = Kσ Aσβ
(β = 1, 2, . . . , p),
(8)
Aασ K σ
(α = 1, 2, . . . , p).
(9)
α
=K A
We formulate and prove several lemmas which are used in the proof of Theorem 2. Lemma 3. Let Mα and Mβ be constants matrices such that max{rank (Mα ), rank (Mβ )} > 1. If for all i ∈ Jα and j ∈ Jβ ∂i Kβσ = hβi (x)Mβ ,
∂j Kασ = hαi (x)Mα , 381
where hβi , hαi are scalar functions of variable x = (x1 , x2 , . . . , xm )T , then ∂i Kβσ = 0,
∂j Kασ = 0
for all i ∈ Jα and j∈ Jβ .
In particular, if K = h(x)M or ∂i K = hi (x)M (i = 1, 2, . . . , m), where M is a constant matrix of rank greater than 1 and h(x), hi (x) are scalar functions, then ∂K = 0. Proof. By the consistency condition (4) in the form ∂j yi = ∂i yj (i, j = 1, 2, . . . , m), for i ∈ Jα and j ∈ Jβ we have hαj Mαi = hβi Mβj , where Mαi and Mβj are columns of the matrices Mα and Mβ respectively. If, for example, hβi = 0 for some i ∈ Jα , then Mβj =
hαj Mαi hβi
for all j ∈ Jβ and, consequently, rank (Mβ ) 1. Since hαj = 0 for all j ∈ Jβ (rank (Mα ) 1), we have Mβj = 0 for all j ∈ Jβ and Mβ = 0. Lemma 4. If n1 > 1, then ∂K1 = 0. Proof. We begin by considering the pth row of the block matrix AK = KA. Since A is quasitriangular, the equality (7) for α = p and β = 1 takes the form App K1p = K1p A11 . Hence K1p is the matrix of an intertwining operator between the irreducible sets Q1 and Qp . If np = n1 , then Q1 and Qp are not equivalent and, by the Schur lemma, K1p = 0 which implies ∂i K p = 0 for all i ∈ J1 in view of the consistency condition (4). If np = n1 , then, by the Schur lemma, the irreducible sets Q1 and Qp are equivalent and the matrix K1p of the intertwining operator is defined up to a factor: K1p = h1 (x)M1p ,
(10)
where h1 (x) is a scalar function of variable x = (x1 , x2 , . . . , xm )T and M1p is a constant nonsingular matrix of order n1 > 1. Hence, by Lemma 3, ∂j K1p = 0 for all j ∈ J1 . For α = p and β = 2 the equality (7) has the form App K2p = K1p A12 + K2p A22 . Applying the differential operator ∂i (i ∈ J1 ) to both sides of this equality, we find App (∂i K2p ) = (∂i K2p ) A22
(i ∈ J1 ).
If np = n2 , then ∂i K2p = 0 for all i ∈ J1 by the Schur lemma. From the consistency condition (4) it follows that ∂j K1p = 0 for all j ∈ J2 . If np = n2 , then ∂i K2p = h2i (x)M2p for all i ∈ J1 by the Schur lemma, where h2i (x) is a scalar function and M2p is a constant nonsingular matrix of order n2 . By (10), from Lemma 3 it follows that ∂i K2p = 0 for all i ∈ J1 and ∂j K1p = 0 for all j ∈ J2 . Applying the operator ∂i (i ∈ J1 ) to both sides of (7) for α = p, β = 3; α = p, β = 4; . . ., α = p, β = p, we find 382
∂i K3p = 0 (i ∈ J1 ),
∂j K1p = 0 (j ∈ J3 ),
∂i K4p = 0 (i ∈ J1 ),
∂j K1p = 0 (j ∈ J4 ),
··· ∂i Kpp = 0 (i ∈ J1 ),
∂j K1p = 0 (j ∈ Jp ).
Thus, for the pth row we have ∂K1p = 0 and ∂i K p = 0 for all i ∈ J1 . These conditions are equivalent because of the consistency condition (4) . Then we consider the (p − 1)th row of the block matrix AK = KA. For α = p − 1 and β = 1 the equality (7) has the form p−1 p p−1 1 + Ap−1 A1 . Ap−1 p K1 = K1 p−1 K1
Applying the operator ∂i (i = 1, 2, . . . , m) to both sides of this equality, we get p−1 ) = (∂i K1p−1 )A11 Ap−1 p−1 (∂i K1
(i = 1, 2, . . . , m).
If np−1 = n1 , then ∂i K1p−1 = 0 (i = 1, 2, . . . , m) by the Schur lemma. By the consistency condition (4), we have ∂j K1p−1 = 0 for all j ∈ J1 . If np−1 = n1 , then (11) ∂i K1p−1 = h1i (x)M1p−1 (i = 1, 2, . . . , m) by the Schur lemma, where h1i (x) is a scalar function and M1p−1 is a constant nonsingular matrix of order n1 > 1. By Lemma 3, ∂i K1p−1 = 0 for all i ∈ J1 . For α = p − 1 and β = 2 the equality (7) takes the form p−1 p p−1 1 + Ap−1 A2 + K2p−1 A22 . Ap−1 p K2 = K1 p−1 K2
Applying the operator ∂i (i ∈ J1 ) to both sides of this equality, we get p−1 ) = (∂i K2p−1 )A22 Ap−1 p−1 (∂i K2
(i ∈ J1 ).
If np−1 = n2 , then, ∂i K2p−1 = 0 for all i ∈ J1 by the Schur lemma. From the consistency condition (4) it follows that ∂j K1p−1 = 0 for all j ∈ J2 . If np−1 = n2 , then ∂i K2p−1 = h2i (x)M2p−1 for all i ∈ J1 by the Schur lemma, where h2i (x) is a scalar function and M2p−1 is a constant nonsingular matrix of order n2 . By (11) and Lemma 3, ∂i K2p−1 = 0 for all i ∈ J1 and ∂j K1p−1 = 0 for all j ∈ J2 . Applying the operator ∂i (i ∈ J1 ) to both sides of (7) with α = p − 1, β = 3; α = p − 1, β = 4; . . .; α = p − 1, β = p, we find ∂i K3p−1 = 0 (i ∈ J1 ),
∂j K1p−1 = 0 (j ∈ J3 ),
∂i K4p−1 = 0 (i ∈ J1 ),
∂j K1p−1 = 0 (j ∈ J4 ),
··· ∂i Kpp−1 = 0 (i ∈ J1 ),
∂j K1p−1 = 0 (j ∈ Jp ).
Thus, for the (p − 1)th row ∂K1p−1 = 0 and ∂i K p−1 = 0 for all i ∈ J1 . Considering the (p − 2)th, (p − 3)th, . . ., 1th rows of AK = KA in a similar way, we find ∂K1σ = 0 (σ = 1, 2, . . . , p) and ∂i K j = 0 for all i ∈ J1 , j = 1, 2, . . . , p. Hence ∂K1 = 0 and ∂i K = 0 for all i ∈ J1 . 383
Lemma 5. If q ∈ (1, p], nq > 1, and ∂Ki = 0 for all i ∈ [1, q − 1], then ∂Kq = 0. Proof. The argument is similar to that in the proof of Lemma 4. Proof of Theorem 2 (continued). The further argument depends on the structure of the composition series (6) of the ring Q. The following alternative holds: either all the quotients Vi /Vi−1 (i = 1, 2, . . . , p) of the composition series (6) have dimension greater than 1, or there exists at least one one-dimensional quotient. 2.1. Multi-dimensional quotients. Suppose that the dimension of every quotient Vi /Vi−1 (i = 1, 2, . . . , p) is greater than 1, i.e., ni > 1 for all i = 1, 2, . . . , p. By Lemmas 4 and 5, the following assertion holds. Proposition 2. If q ∈ [1, p] and n1 > 1, n2 > 1, . . . , nq > 1, then ∂Ki = 0 (i = 1, 2, . . . , q). From Proposition 2 we obtain the conclusion of Theorem 2 in the case where all quotients of the composition series (6) are multi-dimensional. 2.2. One-dimensional quotients. If there is a one-dimensional quotient in (6), then the further consideration depends on the set of weights of the ring Q. Let SpQ be the set of all weights of Q, i.e., the set of complex-valued functions λ = λ(A), A ∈ Q, such that the subspace Vλ = {x ∈ C m : Ax = λ(A)x, A ∈ Q} differs from the zero one. Two cases are possible: (a) SpQ = ∅ or (b) SpQ = ∅. (a) Case SpQ = ∅. In this case, the following assertion holds. Proposition 3. If SpQ = ∅, then ∂K = 0. Proof. Let q be the least natural number such that nq = 1. It is obvious that q > 1. By Proposition 2, ∂Ki = 0 (i = 1, 2, . . . , q − 1). We consider the qth column of the block matrix AK = KA. The equality (8) for β = q takes the form AKq =
q σ=1
Kσ Aσq .
Applying the operator ∂i (i = 1, 2, . . . , m) to both sides of this equality, we find A (∂i Kq ) = (∂i Kq ) Aqq
(i = 1, 2, . . . , m).
Since nq = 1, we have A(∂i Kq ) = Aqq (∂i Kq ) (i = 1, 2, . . . , m). Since the ring Q has no weights, ∂i Kq = 0 (i = 1, 2, . . . , m). If nq+1 > 1, then Lemma 5 implies ∂Kq+1 = 0. If nq+1 = 1, we take into the account that the qth column of the block matrix K is constant and, repeating the above argument for the (q + 1)th column of the block matrix AK = KA, obtain the relation ∂Kq+1 = 0. The constancy of columns of the block matrix K can be proved in a similar way. (b) Case SpQ = ∅. Let SpQ = {λ1 , λ2 , . . . , λk }, and let αi = dim Vλi (i = 1, 2, . . . , k; 1 k p). The composition series (6) can be chosen in such a way that, in the corresponding basis, 384
matrices of a ring, equivalent to the ring Q, have the special quasitriangular form: ⎞ ⎛ 1 A1 A12 . . . A1p ⎜ 0 A2 . . . A2 ⎟ ⎜ p ⎟ 2 A=⎜ .. .. ⎟ ⎟, ⎜ .. . ... . ⎠ ⎝ . 0 0 . . . App where Aij is a matrix with ni rows and nj columns (i, j = 1, 2, . . . , p) such that Aij = 0 for i > j (i, j = 1, 2, . . . , p) and Aii = λi Eαi (i, j = 1, 2, . . . , k), where Eαi is the unit matrix of order αi , ni = αi for i = 1, 2, . . . , k. Moreover, the set Qi = {Aii : A ∈ Q} is irreducible for all i = k + 1, k + 2, . . . , p. To complete the proof of Theorem 2, we need to prove several assertions. Lemma 6. Let SpQ = {λ1 , λ2 , . . . , λk } (1 k p). If Q has no (r, l)-pair, then ∂K1 = 0. Proof. The commutability condition (3) expressed in terms of columns of the block matrix K as the equality (8) takes the following form for β = 1: AK1 = K1 A11 = λ1 (A)K1 . This means that K1 ∈ Vλ1 . By the structure of the matrix A and the irreducibility of the set Qi = {Aii : A ∈ Q} for every i = k + 1, k + 2, . . . , p, we have K1σ = 0 for all σ > α1 , which implies ∂1 K σ = 0
for all σ > α1
(12)
in view of the consistency condition (4). The commutability condition (3) expressed in terms of rows of the block matrix K as the equality (9) takes the following form for α = α1 : K α1 A =
p
α
Aσ 1 K σ .
σ=α1
Applying the operator ∂1 to both sides of the last equality, we find α
(∂1 K α1 )A = Aα11 (∂1 K α1 ) = λ1 (A)(∂1 K α1 ). Since the ring Q has no (r, l)-pair, we have ∂1 K α1 = 0. For α = α1 − 1 the equality (9) has the form K
α1 −1
A=
p
α −1
Aσ 1
Kσ.
σ=α1 −1
Applying the operator ∂1 to both sides of this equality, we get α −1
(∂1 K α1 −1 )A = Aα1 −1 (∂1 K α1 −1 ) = λ1 (A)(∂1 K α1 −1 ). 1
Since the ring Q has no (r, l)-pair, ∂1 K α1 −1 = 0. Considering the commutability condition in terms of rows for α = α1 − 2, α = α1 − 3, . . ., α = 1, we obtain the equalities ∂1 K α1 −2 = 0, ∂1 K α1 −3 = 0, . . . , ∂1 K 1 = 0. Thus, ∂1 K σ = 0 for all σ α1 . Together with (12), this implies ∂1 K = 0. From the consistency condition (4) it follows that ∂K1 = 0. 385
Lemma 7. Suppose that q ∈ (1, p] and nq > 1. If ∂Ki = 0 for all i ∈ [1, q − 1] and the ring Q has no (r, l)-pair, then ∂Kq = 0. Proof. For β = q the equality (8) has the form AKq =
q σ=1
Kσ Aσq .
Applying the operator ∂i (i = 1, 2, . . . , m) to both sides of the last equality, we get A(∂i Kq ) = (∂i Kq )Aqq = Aqq (∂i Kq )
(i = 1, 2, . . . , m).
If λ : A → Aqq is not a weight of the ring Q, then ∂i Kq = 0 (i = 1, 2, . . . , m). If λ : A → Aqq is a weight of the ring Q, i.e., Aqq = λs (A) for some s ∈ [1, k], then ∂i Kq ∈ Vλs (i = 1, 2, . . . , m). The further argument is the same as that in the proof of Lemma 6. Lemmas 6 and 7 imply the following assertion. Proposition 4. If a ring Q has no (r, l)-pair and the set of weights of Q is not empty, then ∂Ki = 0 (i = 1, 2, . . . , q) for any q ∈ (1, p] such that nq = 1. Proof of Theorem 2 (continued). From Proposition 4 we obtain the conclusion of Theorem 2 in the case SpQ = ∅. Indeed, let q be the least natural number such that nq = 1. By Proposition 4, ∂Ki = 0 (i = 1, 2, . . . , q). If nq+1 > 1, then ∂Kq+1 = 0 by Lemma 5. If nq+1 = 1, then ∂Kq+1 = 0 by Proposition 4. Considering nq+2 , nq+3 , . . . , np in a similar way, we obtain the equalities ∂Kq+2 = 0, ∂Kq+3 = 0, . . . , ∂Kp = 0. Hence ∂K = 0. Theorem 2 is proved. Corollary. If a ring of matrices has no (r, l)-pair, then the Jacobi matrix which commutes with all matrices of this ring is constant.
Acknowledgments The work is financially supported by the Russian Foundation for Basic Research (project No. 10-01-00497) and the Integration project of the Siberian Branch of the Russian Academy of Sciences (project No. 103).
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Submitted on June 9, 2009 386
