Perturbation analysis of a matrix differential equation = ABx

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Abstract

Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + , B + ) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of and . Each perturbation (, ) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + )(B + ) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

Abstract

Two complex matrix pairs (A, B) and (A′, B′) are contragrediently equivalent if there are nonsingular S and R such that (A′, B′) = (S−1AR, R−1BS). M.I. García-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair (A, B) for contragredient equivalence; that is, a simple normal form to which all matrix pairs (A + , B + ) close to (A, B) can be reduced by contragredient equivalence transformations that smoothly depend on the entries of and . Each perturbation (, ) of (A, B) defines the first order induced perturbation AB͠ + A͠B of the matrix AB, which is the first order summand in the product (A + )(B + ) = AB + AB͠ + A͠B + A͠B͠. We find all canonical matrix pairs (A, B), for which the first order induced perturbations AB͠ + A͠B are nonzero for all nonzero perturbations in the normal form of García-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations = Cx, whose product of two matrices: C = AB; using the substitution x = Sy, one can reduce C by similarity transformations S−1CS and (A, B) by contragredient equivalence transformations (S−1AR, R−1BS).

1 Introduction

We study a matrix differential equation = ABx, whose matrix is a product of an m × n complex matrix A and an n × m complex matrix B. It is equivalent to = S−1ARR−1BSy, in which S and R are nonsingular matrices and x = Sy. Thus, we can reduce (A, B) by transformations of contragredient equivalence

(A,B)(S1AR,R1BS),S and R are nonsingular.

The canonical form of (A, B) with respect to these transformations was obtained by Dobrovol′skaya and Ponomarev [3] and, independently, by Horn and Merino [5]:

each pair(A,B)is contragrediently equivalent to a direct sum, uniquely determined up topermutation of summands, of pairs of the types(Ir,Jr(λ)),(Jr(0),Ir),(Fr,Gr),(Gr,Fr),

in which r = 1, 2, …,

Jr(λ):=λ10λ10λ(λC),Fr:=001001,Gr:=100001

are r × r, r × (r − 1), (r − 1) × r matrices, and

(A1,B1)(A2,B2):=(A1A2,B1B2).

Note that (F1, G1) = (010, 010); we denote by 0mn the zero matrix of size m × n, where m, n ∈ {0, 1, 2, …}. All matrices that we consider are complex matrices. All matrix pairs that we consider are counter pairs: a matrix pair (A, B) is a counter pair if A and BT have the same size.

A notion of miniversal deformation was introduced by Arnold [1, 2]. He constructed a miniversal deformation of a Jordan matrix J; i.e., a simple normal form to which all matrices J + E close to J can be reduced by similarity transformations that smoothly depend on the entries of E. García-Planas and Sergeichuk [4] constructed a miniversal deformation of a canonical pair (2) for contragredient equivalence (1).

For a counter matrix pair (A, B), we consider all matrix pairs (A + , B + ) that are sufficiently close to (A, B). The pair (, ) is called a perturbation of (A, B). Each perturbation (, ) of (A, B) defines the induced perturbation AB͠ + A͠B + A͠B͠ of the matrix AB that is obtained as follows:

(A+A~)(B+B~)=AB+AB~+A~B+A~B~.

Since and are small, their product A͠B͠ is “very small”; we ignore it and consider only first order induced perturbations AB͠ + A͠B of AB.

In this paper, we describe all canonical matrix pairs (A, B) of the form (2), for which the first order induced perturbations AB͠ + A͠B are nonzero for all miniversal perturbations (, ) ≠ 0 in the normal form defined in [4].

Note that z = ABx can be considered as the superposition of the systems y = Bx and z = Ay:

xByAzimpliesxABz

2 Miniversal deformations of counter matrix pairs

In this section, we recall the miniversal deformations of canonical pairs (2) for contragredient equivalence constructed by García-Planas and Sergeichuk [4].

Let

(A,B)=(I,C)j=1t1(Ir1j,Jr1j)j=1t2(Jr2j,Ir2j)j=1t3(Fr3j,Gr3j)j=1t4(Gr4j,Fr4j)

be a canonical pair for contragredient equivalence, in which

C:=i=1tΦ(λi),Φ(λi):=Jmi1(λi)Jmiki(λi)withλiλjifij,

mi1mi2 ⩾ … ⩾ miki, and ri1ri2 ⩾ … ⩾ riti.

For each matrix pair (A, B) of the for (3), we define the matrix pair

(I,iΦ(λi)+N))jIr1j000jJr2j(0)+NN0NP3N0Q4,jJr1j(0)+NNNNjIr2j0N0Q30NP4,

of the same size and of the same partition of the blocks, in which

N:=[Hij]

is a parameter block matrix with pi × qj blocks Hij of the form

Hij:=0ifpiqj,Hij:=0ifpi>qj.

Pl:=Frl1+HHHFrl2+HH0Frltl+H,Ql:=Grl10HGrl2HHGrltl(l=3,4),

N and H are matrices of the form (5) and (6), and the stars denote independent parameters.

Theorem 1

(see [4]). Let (A, B) be the canonical pair (3). Then all matrix pairs (A + , B + ) that are sufficiently close to (A, B) are simultaneously reduced by some transformation

(A+A~,B+B~)(S1(A+A~)R,R1(B+B~)S),

in which S and R are matrix functions that depend holomorphically on the entries of and , S(0) = I, and R(0) = I, to the form (4), whose stars are replaced by complex numbers that depend holomorphically on the entries of and . The number of stars is minimal that can be achieved by such transformations.

3 Main theorem

Each matrix pair (A + , B + ) of the form (4), in which the stars are complex numbers, we call a miniversal normal pair and (, ) a miniversal perturbation of (A, B).

The following theorem is the main result of the paper.

Theorem 2

Let (A, B) be a canonical pair (2). The following two conditions are equivalent:

  • AB͠ + A͠B ≠ 0 for all nonzero miniversal perturbations (, ).

  • (A, B) does not contain

    • (Ir, Jr(0)) ⊕ (Jr(0), Ir) for each r,

    • (F1, G1) ⊕ (G2, F2), and

    • (Fm, Gm) ⊕ (Gm, Fm) for each m.

Proof

(a) ⟹ (b). Let (A, B) be a canonical pair (2). We should prove that if (A, B) contains a pair of type (i), (ii), or (iii), then AB͠ + A͠B = 0 for some miniversal perturbation (, ) ≠ (0, 0). It is sufficient to prove this statement for (A, B) of types (i)–(iii).

  • (A, B) = (Ir, Jr(0)) ⊕ (Jr(0), Ir) for some r. We should prove that there exists a nonzero miniversal perturbation (, ) such that AB͠ + A͠B = 0.

    If r = 1, then

    (A,B)=(I1,J1(0))(J1(0),I1)=1000,0001.

    Its miniversal deformation (4) has the form

    100ε,λμδ1,

    in which ε, λ, μ and δ are independent parameters. We have that

    AB~+A~B=000ε+λμ00=λμ0ε.

    Choosing ε = μ = λ = 0 and δ ≠ 0, we get A͠B + B͠A = 0.

    If r = 2, then (A, B) = (I2, J2(0)) ⊕ (J2(0), I2) and

    (A+A~,B+B~)=10000100000100ε7ε8,0100ε1ε2ε3ε4ε5010ε6001,

    We get

    AB~+A~B=0000ε1ε2ε3ε4ε60000000+00000000000000ε7ε8=0000ε1ε2ε3ε4ε600000ε7ε8.

    Choosing ε5 ≠ 0 and εi = 0 if i ≠ 5, we get AB͠ + A͠B = 0.

    If r is arbitrary, then (A, B) = (Ir, Jr(0)) ⊕ (Jr(0), Ir) and its miniversal deformation has the form

    110101001α1α2αs,01001ε1ε2εrεr+1εr+2εr+sβ11β210βs1,

    in which all αi, βi, εi are independent parameters. Taking all parameters zero except for β1 ≠ 0, we get that AB͠ + A͠B = 0.

  • (A, B) = (F1, G1) ⊕ (G2, F2). Then

    (A+A~,B+B~)=εδ01,01λμ,

    in which ε, δ, λ and μ are independent parameters. We get

    AB~+A~B=0ε00+00λμ=0ελμ.

    Taking all parameters zero except for δ ≠ 0, we get that AB͠ + A͠B = 0.

  • (A, B) = (Fm, Gm) ⊕ (Gm, Fm) for some m.

    If m = 1, then (A, B) = (F1, G1) ⊕ (G1, F1) = (0, 0). For each perturbation (, ) ≠ (0, 0), we get AB͠ + A͠B = 0.

    If m = 2, then the miniversal deformation (4) of (A, B) is

    (A+A~,B+B~)=1α0εβ0001,010001λμδ

    in which ε, α, β, λ, μ and δ are independent parameters. We obtain

    AB~+A~B=000000λμδ+0000εβ000=0000εβλμδ.

    Choosing all parameters zero except for α ≠ 0, we get AB͠ + A͠B = 0.

    If r is arbitrary, then the miniversal deformation (4) of (A, B) has the form

    10εr001ε2r2ε1εr1ε2r10100001,010000110001α1α2αrαr+1α2r1

    in which all αi and εj are independent parameters. Since the rth row of B is zero, a parameter ε2r−2 does not appear in B, and so in AB͠ + A͠B too. Choosing all parameters zeros except for ε2r−2 ≠ 0, we get AB͠ + A͠B = 0.

    (b) ⟹ (a). Let us prove that if there exists a nonzero miniversal perturbation (, ) such that AB͠ + A͠B = 0, then (A, B) contains (Ir, Jr(0)) ⊕ (Jr(0), Ir) for some r, or (F1, G1) ⊕ (G2, F2), or (Fm, Gm) ⊕ (Gm, Fm) for some m.

    Since the deformation (4) is the direct sum of

    (I,i(Φ(λi)+N))andjIr1j000jJr2j(0)+NN0NP3N0Q4,jJr1j(0)+NNNNjIr2j0N0Q30NP4,

    it is sufficient to consider (A, B) equals

    (I,i(Φ(λi)))orj=1t1(Ir1j,Jr1j)j=1t2(Jr2j,Ir2j)j=1t3(Fr3j,Gr3j)j=1t4(Gr4j,Fr4j).

    Let first (A, B) = (I, ⨁i(Φ(λi))). Then

    (A+A~,B+B~)=jIr1j000000jIrlj,jJr1j(λ1)+N000000jJrlj(λl)+N.

    If

    A~B+A~B=N000000N=0,

    in which all N have independent parameters, then all N are zero, and so (, ) = (0, 0).

    It remains to consider (A, B) equaling the second pair in (8). Write the matrices (7) as follows:

    Pl=P¯l+P_l,Ql=Q¯l+Q_l, in which l=3,4,

    P¯l=Frl100Frl200Frltl,P_l=Hrl1HHHrl2H0Hrltl,Q¯l=Grl100Grl200Grltl,Q_l=0rl10H0rl2HH0rltl,

    N and H are matrices of the form (5) and(6), and the stars denote independent parameters.

    Write

    J1:=jJr1j(0),J2:=jJr2j(0).

    Then

    A=I0000J20000P¯30000Q¯4,A~=00000NNN0NP_3N0N0Q_4,B=J10000I0000Q¯30000P¯4,B~=NNNNN000N0Q_30N0NP_4,AB~=NNNNJ2N000P¯3N0P¯3Q_30Q¯4N0Q¯4NQ¯4P_4,A~B=00000NNQ¯3NP¯40NP_3Q¯3NP¯40N0Q_4P¯4,

    in which we denote by N blocks of the form (5). All blocks denoted by N have distinct sets of independent parameters and may have distinct sizes.

    Since A͠B and AB͠ have independent parameters for each (A, B), we should prove that A͠B ≠ 0 for all ≠ 0 and A ≠ 0 for all ≠ 0. Thus, we should prove that

    J2N,NP¯4,P¯3N,NQ¯3,Q¯4N

    are nonzero if the corresponding parameter blocks N are nonzero.

    Let us consider the first matrix in (10):

    J2N=Jr10Jr20JrnHr10Hr20Hrn=0ε11ε1m1000εn1εnmn00,

    in which all εij are independent parameters and r1r2 ⩽ … ⩽ rn. Clearly, J2 N ≠ 0 if at least one εij ≠ 0.

    Let us consider the second matrix in (10):

    NP¯4=Hr10Hr20HrnFr10Fr20Frn=0ε11ε1m10εn1εnmn.

    in which all εj are independent parameters and r1r2 ⩾ … ⩾ rn. Clearly, N P4 ≠ 0 if at least one εij ≠ 0.

    The matrices P3N, Q4N, N Q3, and Q4N in (10) are considered analogously.

Communicated by Juan L.G. Guirao

References

  • [1]

    Arnold, V. I. (1971), On matrices depending on parameters, Russian Math. Surveys 26, pp. 29–43,

  • [2]

    Arnold, V. I. (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag,

  • [3]

    Dobrovol’skaya, N. M. and Ponomarev, V. A. (1965), A pair of counter-operators (in Russian), Uspehi Mat. Nauk 20, pp. 80–86,

  • [4]

    Garcia-Planas, M. I. and Sergeichuk, V. V. (1999), Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils, Linear Algebra Appl. 302/303, pp. 45–61,

  • [5]

    Horn, R. A. and Merino, D. I. (1995), Contragredient equivalence: A canonical form and some applications, Linear Algebra Appl. 214, pp. 43–92,

[1]

Arnold, V. I. (1971), On matrices depending on parameters, Russian Math. Surveys 26, pp. 29–43,

[2]

Arnold, V. I. (1988), Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag,

[3]

Dobrovol’skaya, N. M. and Ponomarev, V. A. (1965), A pair of counter-operators (in Russian), Uspehi Mat. Nauk 20, pp. 80–86,

[4]

Garcia-Planas, M. I. and Sergeichuk, V. V. (1999), Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils, Linear Algebra Appl. 302/303, pp. 45–61,

[5]

Horn, R. A. and Merino, D. I. (1995), Contragredient equivalence: A canonical form and some applications, Linear Algebra Appl. 214, pp. 43–92,

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