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Introduction
We study a matrix differential equation ẋ = ABx, whose matrix is a product of an m × n complex matrix A and an n × m complex matrix B. It is equivalent to ẏ = 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
$$\begin{array}{}
\displaystyle
(A,B)\mapsto
(S^{-1}AR,R^{-1}BS), \qquad S \text{ and } R \text{ are nonsingular.}
\end{array}$$
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]:
$$\begin{array}{}
\displaystyle
\text {each
pair}~~ (A,B)~~ \text{is contragrediently equivalent to
a direct sum, uniquely determined up to}\\ \text{permutation of summands, of pairs of the types}~~
(I_r,J_r(\lambda)),\,
(J_r(0),I_r), \,
(F_r,G_{r}),\,
(G_{r},F_{r}),
\end{array}$$
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 + A͠, B + B͠) that are sufficiently close to (A, B). The pair (A͠, B͠) is called a perturbation of (A, B). Each perturbation (A͠, B͠) of (A, B) defines the induced perturbationAB͠ + A͠B + A͠B͠ of the matrix AB that is obtained as follows:
$$\begin{array}{}
\displaystyle
(A +\widetilde{A})(B+\widetilde{B}) = AB + A\widetilde{B}+\widetilde{A}B + \widetilde{A}\widetilde{B}.
\end{array}$$
Since A͠ and B͠ are small, their product A͠B͠ is “very small”; we ignore it and consider only first order induced perturbationsAB͠ + 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 (A͠, B͠) ≠ 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:
$$\begin{array}{}
\displaystyle
x \longrightarrow \boxed{B} \xrightarrow{y} \boxed{A} \longrightarrow z \qquad \text{implies} \qquad x \longrightarrow \boxed{AB}\longrightarrow z
\end{array}$$
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].
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 + A͠, B + B͠) that are sufficiently close to (A, B) are simultaneously reduced by some transformation
$$\begin{array}{}
\displaystyle
(A+\widetilde A, B+\widetilde B) \mapsto (S^{-1}(A+\widetilde A)R,R^{-1}(B+\widetilde B)S),
\end{array}$$
in whichSandRare matrix functions that depend holomorphically on the entries ofA͠andB͠, S(0) = I, and R(0) = I, to the form(4), whose stars are replaced by complex numbers that depend holomorphically on the entries ofA͠andB͠. The number of stars is minimal that can be achieved by such transformations.
Main theorem
Each matrix pair (A + A͠, B + B͠) of the form (4), in which the stars are complex numbers, we call a miniversal normal pair and (A͠, B͠) 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 (Ã, B̃).
(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 (A͠, B͠) ≠ (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 (A͠, B͠) such that AB͠ + A͠B = 0.
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 A͠B, 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 (Ã, B̃) 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.
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 A͠ ≠ 0 and B͠A ≠ 0 for all B͠ ≠ 0. Thus, we should prove that