Perturbation analysis of a matrix differential equation  =

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

$\begin{matrix} (A, B) \mapsto (S^{- 1} A R, R^{- 1} B S), S and R are nonsingular. \end{matrix}$ $$\begin{array}{} \displaystyle (A,B)\mapsto (S^{-1}AR,R^{-1}BS), \qquad S \text{ and } R \text{ are nonsingular.} \end{array}$$(1)

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]:

$\begin{matrix} each pair (A, B) is contragrediently equivalent to a direct sum, uniquely determined up to \\ permutation of summands, of pairs of the types (I_{r}, J_{r} (λ)), (J_{r} (0), I_{r}), (F_{r}, G_{r}), (G_{r}, F_{r}), \end{matrix}$ $$\begin{array}{} \displaystyle \text {each pair}~~ (A,B)~~ \text{is contragrediently equivalent to a direct sum, uniquely determined up to}\\ \text{permutation of summands, of pairs of the types}~~ (I_r,J_r(\lambda)),\, (J_r(0),I_r), \, (F_r,G_{r}),\, (G_{r},F_{r}), \end{array}$$(2)

in which r = 1, 2, …,

$\begin{matrix} J_{r} (λ) := (\begin{matrix} λ & 1 & 0 \\ λ & ⋱ \\ ⋱ & 1 \\ 0 & λ \end{matrix}) (λ \in C), F_{r} := (\begin{matrix} 0 & 0 \\ 1 & ⋱ \\ ⋱ & 0 \\ 0 & 1 \end{matrix}), G_{r} := (\begin{matrix} 1 & 0 & 0 \\ ⋱ & ⋱ \\ 0 & 0 & 1 \end{matrix}) \end{matrix}$ $$\begin{array}{} \displaystyle J_r(\lambda):=\begin{bmatrix} \lambda&1&&0\\[-2mm] &\lambda&\ddots&\\[-2mm] &&\ddots&1\\ 0&&&\lambda \end{bmatrix} \ (\lambda \in \mathbb C), \quad F_r:=\begin{bmatrix} 0&&0\\[-6pt]1&\ddots&\\[-6pt]&\ddots&0\\0&&1 \end{bmatrix}, \quad G_r:=\begin{bmatrix} 1&0&&0\\[-6pt]&\ddots&\ddots\\[-3pt]0&&0&1 \end{bmatrix} \end{array}$$

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

$\begin{matrix} (A_{1}, B_{1}) \oplus (A_{2}, B_{2}) := (A_{1} \oplus A_{2}, B_{1} \oplus B_{2}) . \end{matrix}$ $$\begin{array}{} \displaystyle (A_1,B_1) \oplus (A_2,B_2):=(A_1 \oplus A_2, B_1 \oplus B_2). \end{array}$$

Note that (F₁, G₁) = (0₁₀, 0₁₀); we denote by 0_mn 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 B^T 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:

$\begin{matrix} (A + \tilde{A}) (B + \tilde{B}) = A B + A \tilde{B} + \tilde{A} B + \tilde{A} \tilde{B} . \end{matrix}$ $$\begin{array}{} \displaystyle (A +\widetilde{A})(B+\widetilde{B}) = AB + A\widetilde{B}+\widetilde{A}B + \widetilde{A}\widetilde{B}. \end{array}$$

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:

$\begin{matrix} x ⟶ B \overset{y}{\to} A ⟶ z implies x ⟶ A B ⟶ z \end{matrix}$ $$\begin{array}{} \displaystyle x \longrightarrow \boxed{B} \xrightarrow{y} \boxed{A} \longrightarrow z \qquad \text{implies} \qquad x \longrightarrow \boxed{AB}\longrightarrow z \end{array}$$

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].

Let

$\begin{matrix} (A, B) = (I, C) \oplus ⨁_{j = 1}^{t_{1}} (I_{r_{1 j}}, J_{r_{1 j}}) \oplus ⨁_{j = 1}^{t_{2}} (J_{r_{2 j}}, I_{r_{2 j}}) \oplus ⨁_{j = 1}^{t_{3}} (F_{r_{3 j}}, G_{r_{3 j}}) \oplus ⨁_{j = 1}^{t_{4}} (G_{r_{4 j}}, F_{r_{4 j}}) \end{matrix}$ $$\begin{array}{} \displaystyle (A,B)=(I,C)\oplus \bigoplus_{j=1}^{t_1} (I_{r_{1j}},J_{r_{1j}}) \oplus\bigoplus_{j=1}^{t_2}(J_{r_{2j}},I_{r_{2j}}) \oplus\bigoplus_{j=1}^{t_3}(F_{r_{3j}},G_{r_{3j}}) \oplus\bigoplus_{j=1}^{t_4}(G_{r_{4j}},F_{r_{4j}}) \end{array}$$(3)

be a canonical pair for contragredient equivalence, in which

$\begin{matrix} C := ⨁_{i = 1}^{t} Φ (λ_{i}), Φ (λ_{i}) := J_{m_{i 1}} (λ_{i}) \oplus \dots \oplus J_{m_{i k_{i}}} (λ_{i}) with λ_{i} \neq λ_{j} if i \neq j, \end{matrix}$ $$\begin{array}{} \displaystyle C := \bigoplus_{i=1}^t \Phi({\lambda_i}), \qquad \Phi({\lambda_i}):=J_{m_{i1}}(\lambda_i)\oplus \dots \oplus J_{m_{ik_i}}(\lambda_i) \qquad \text{with}~~ \lambda_i \neq \lambda_j ~~\text{if}~~ i \neq j, \end{array}$$

m_i1 ⩾ m_i2 ⩾ … ⩾ m_{ik_i}, and r_i1 ⩾ r_i2 ⩾ … ⩾ r_{it_i}.

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

$\begin{matrix} (I, ⨁_{i} Φ (λ_{i}) + N)) \oplus ((\begin{matrix} \oplus_{j} I_{r_{1 j}} & 0 & 0 \\ 0 & \oplus_{j} J_{r_{2 j}} (0) + N & N \\ 0 & N & \begin{matrix} P_{3} & N \\ 0 & Q_{4} \end{matrix} \end{matrix}), (\begin{matrix} \oplus_{j} J_{r_{1 j}} (0) + N & N & N \\ N & \oplus_{j} I_{r_{2 j}} & 0 \\ N & 0 & \begin{matrix} Q_{3} & 0 \\ N & P_{4} \end{matrix} \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle \Big(I,\bigoplus_i\Phi({\lambda_i})+{N})\Big) \oplus \left(\left[ \begin{array}{c|c|c} \oplus_j I_{r_{1j}}&0&0 \\\hline 0&\oplus_j J_{r_{2j}}(0)+N&N \\ \hline 0&N& \begin{matrix} P_3&N\\0&Q_4 \end{matrix} \end{array}\right], \left[\begin{array}{c|c|c} \oplus_j J_{r_{1j}}(0)+N &N & N\\ \hline N & \oplus_j I_{r_{2j}} &0\\ \hline N &0&\begin{matrix} Q_3&0\\ N&P_4 \end{matrix} \end{array}\right]\right), \end{array}$$(4)

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

$\begin{matrix} N := [H_{i j}] \end{matrix}$ $$\begin{array}{} \displaystyle {N}:=[H_{ij}] \end{array}$$(5)

is a parameter block matrix with p_i × q_j blocks H_ij of the form

$\begin{matrix} H_{i j} := (\begin{matrix} * \\ ⋮ & 0 \\ * \end{matrix}) i f p_{i} \leq q_{j}, H_{i j :} = (\begin{matrix} 0 \\ * \dots * \end{matrix}) i f p_{i} > q_{j} . \end{matrix}$ $$\begin{array}{} \displaystyle H_{ij}:=\left[ \begin{matrix}*& \\ \vdots&\!\!\!\Large 0\\ *& \end{matrix} \right] ~~ {\rm if}~~ p_i \le q_j, \qquad H_{ij:}=\left[\begin{matrix} \Large 0 \\ \!\!* \cdots *\!\! \end{matrix}\right] ~~ {\rm if}~~ p_i \gt q_j. \end{array}$$(6)

$\begin{matrix} P_{l} := (\begin{matrix} F_{r_{l 1}} + H & H & \dots & H \\ F_{r_{l 2}} + H & ⋱ & ⋮ \\ ⋱ & H \\ 0 & F_{r_{l t_{l}}} + H \end{matrix}), & Q_{l} := (\begin{matrix} G_{r_{l 1}} & 0 \\ H & G_{r_{l 2}} \\ ⋮ & ⋱ & ⋱ \\ H & \dots & H & G_{r_{l t_{l}}} \end{matrix}) & (l = 3, 4), \end{matrix}$ $$\begin{array}{} \displaystyle P_l:=\left[\begin{matrix} F_{r_{l1}}+H &H&\cdots &H\\ &F_{r_{l2}}+H&\ddots &\vdots\\ &&\ddots &H\\ 0&&& F_{r_{lt_l}}+H \end{matrix}\right],&&& Q_l:=\left[\begin{matrix} G_{r_{l1}}&&& 0 \\ H&G_{r_{l2}}&& \\ \vdots&\ddots&\ddots&\\ H&\cdots&H&G_{r_{lt_l}} \end{matrix}\right]&&&(l=3,~4), \end{array}$$(7)

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

$\begin{matrix} (A + \tilde{A}, B + \tilde{B}) \mapsto (S^{- 1} (A + \tilde{A}) R, R^{- 1} (B + \tilde{B}) S), \end{matrix}$ $$\begin{array}{} \displaystyle (A+\widetilde A, B+\widetilde B) \mapsto (S^{-1}(A+\widetilde A)R,R^{-1}(B+\widetilde B)S), \end{array}$$

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

(I_r, J_r(0)) ⊕ (J_r(0), I_r) for each r,

(F₁, G₁) ⊕ (G₂, F₂), and

(F_m, G_m) ⊕ (G_m, F_m) 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).

Case 1

(A, B) = (I_r, J_r(0)) ⊕ (J_r(0), I_r) for some r. We should prove that there exists a nonzero miniversal perturbation (A͠, B͠) such that AB͠ + A͠B = 0.

If r = 1, then

$\begin{matrix} (A, B) = (I_{1}, J_{1} (0)) \oplus (J_{1} (0), I_{1}) = ((\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}), (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix})) . \end{matrix}$ $$\begin{array}{} \displaystyle (A,B)=(I_1, J_1(0))\oplus (J_1(0),I_1) =\left( \left[ \begin{array}{c|c} 1&0\\ \hline 0&0\\ \end{array}\right], \left[ \begin{array}{c|c} 0&0\\ \hline 0&1\\ \end{array} \right] \right). \end{array}$$

Its miniversal deformation (4) has the form

$\begin{matrix} ((\begin{matrix} 1 & 0 \\ 0 & ε \end{matrix}), (\begin{matrix} λ & μ \\ δ & 1 \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle \left( \left[ \begin{array}{c|c} 1&0\\ \hline 0&\varepsilon\\ \end{array}\right], \left[ \begin{array}{c|c} \lambda&\mu\\ \hline \delta&1\\ \end{array} \right] \right), \end{array}$$

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

$\begin{matrix} A \tilde{B} + \tilde{A} B = (\begin{matrix} 0 & 0 \\ 0 & ε \end{matrix}) + (\begin{matrix} λ & μ \\ 0 & 0 \end{matrix}) = (\begin{matrix} λ & μ \\ 0 & ε \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle A \widetilde B + \widetilde AB= \left[ \begin{array}{c|c} 0&0\\ \hline 0&\varepsilon\\ \end{array}\right] + \left[ \begin{array}{c|c} \lambda&\mu\\ \hline 0&0\\ \end{array} \right] = \left[ \begin{array}{c|c} \lambda&\mu\\ \hline 0&\varepsilon\\ \end{array} \right]. \end{array}$$

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

$\begin{matrix} (A + \tilde{A}, B + \tilde{B}) = ((\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & ε_{7} & ε_{8} \end{matrix}), (\begin{matrix} 0 & 1 & 0 & 0 \\ ε_{1} & ε_{2} & ε_{3} & ε_{4} \\ ε_{5} & 0 & 1 & 0 \\ ε_{6} & 0 & 0 & 1 \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle (A+\widetilde A,B+\widetilde B)= \left( \left[ \begin{array}{cc|cc} 1&0&0&0\\ 0&1&0&0\\ \hline 0&0&0&1 \\ 0&0&\varepsilon_7&\varepsilon_8 \\ \end{array}\right], \left[ \begin{array}{cc|cc} 0&1&0&0\\ \varepsilon_1&\varepsilon_2 & \varepsilon_3&\varepsilon_4\\ \hline \varepsilon_5&0&1&0 \\ \varepsilon_6&0&0&1 \\ \end{array} \right] \right), \end{array}$$

We get

$\begin{matrix} A \tilde{B} + \tilde{A} B = (\begin{matrix} 0 & 0 & 0 & 0 \\ ε_{1} & ε_{2} & ε_{3} & ε_{4} \\ ε_{6} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) + (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & ε_{7} & ε_{8} \end{matrix}) = (\begin{matrix} 0 & 0 & 0 & 0 \\ ε_{1} & ε_{2} & ε_{3} & ε_{4} \\ ε_{6} & 0 & 0 & 0 \\ 0 & 0 & ε_{7} & ε_{8} \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle A\widetilde{B} +\widetilde{A}B= \begin{bmatrix} 0&0&0&0\\ \varepsilon_1&\varepsilon_2 & \varepsilon_3&\varepsilon_4\\ \varepsilon_6& 0&0 &0\\ 0&0&0&0 \end{bmatrix}+ \begin{bmatrix} 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&\varepsilon_7&\varepsilon_8 \end{bmatrix} = \begin{bmatrix} 0&0&0&0\\ \varepsilon_1&\varepsilon_2 & \varepsilon_3&\varepsilon_4\\ \varepsilon_6&0 &0 &0\\ 0&0&\varepsilon_7&\varepsilon_8 \end{bmatrix}. \end{array}$$

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

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

$\begin{matrix} ((\begin{matrix} 1 \\ 1 \\ ⋱ & 0 \\ 1 \\ 0 & 1 \\ ⋱ & ⋱ \\ 0 & 0 & 1 \\ α_{1} & α_{2} & \dots & α_{s} \end{matrix}), (\begin{matrix} 0 & 1 \\ ⋱ & ⋱ & 0 \\ 0 & 1 \\ ε_{1} & ε_{2} & \dots & ε_{r} & ε_{r + 1} & ε_{r + 2} & \dots & ε_{r + s} \\ β_{1} & 1 \\ β_{2} & 1 \\ ⋮ & 0 & ⋱ \\ β_{s} & 1 \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle \left( \left[ \begin{array}{cccc|cccc} 1&&&&&\\[-3 pt] &1&&&&&\\ &&\ddots&&&&{0}\\ & &&1&&&\\ \hline &&&&0&1&& \\ &&&&&\ddots&\ddots& \\ &&\text{0}&&&&0&1 \\ &&&&\alpha_1&\alpha_2&\dots&\alpha_{s}\\ \end{array}\right], \left[ \begin{array}{cccc|cccc} 0&1&&&&\\ &\ddots&\ddots&&&&\!\!\!\!\!\!\!\!\!0\\ &&0&1&&&\\ \varepsilon_1&\varepsilon_2 &\dots&\varepsilon_r&\varepsilon_{r+1}&\varepsilon_{r+2}&\dots&\varepsilon_{r+s} \\ \hline \beta_1&&&&1&&& \\ \beta_2&&&&&1&& \\ \vdots&&\text{0}&&&&\ddots& \\ \beta_s &&&&&&&1\\ \end{array} \right] \right), \end{array}$$

in which all α_i, β_i, ε_i are independent parameters. Taking all parameters zero except for β₁ ≠ 0, we get that AB͠ + A͠B = 0.

Case 2

(A, B) = (F₁, G₁) ⊕ (G₂, F₂). Then

$\begin{matrix} (A + \tilde{A}, B + \tilde{B}) = ((\begin{matrix} ε & δ \\ 0 & 1 \end{matrix}), (\begin{matrix} 0 & 1 \\ λ & μ \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle (A+\widetilde A, B+ \widetilde B) = \left( \left[ \begin{matrix} \varepsilon&\delta\\ 0&1\\ \end{matrix}\right], \left[ \begin{matrix} 0&1 \\ \lambda&\mu \\ \end{matrix}\right] \right), \end{array}$$

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

$\begin{matrix} A \tilde{B} + \tilde{A} B = (\begin{matrix} 0 & ε \\ 0 & 0 \end{matrix}) + (\begin{matrix} 0 & 0 \\ λ & μ \end{matrix}) = (\begin{matrix} 0 & ε \\ λ & μ \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle A\widetilde{B} +\widetilde{A}B= \left[ \begin{matrix} 0&\varepsilon\\ 0&0\\ \end{matrix}\right]+ \left[ \begin{matrix} 0&0\\ \lambda&\mu\\ \end{matrix}\right] = \left[ \begin{matrix} 0&\varepsilon\\ \lambda&\mu\\ \end{matrix}\right]. \end{array}$$

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

Case 3

(A, B) = (F_m, G_m) ⊕ (G_m, F_m) for some m.

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

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

$\begin{matrix} (A + \tilde{A}, B + \tilde{B}) = ((\begin{matrix} 1 & α & 0 \\ ε & β & 0 \\ 0 & 0 & 1 \end{matrix}), (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ λ & μ & δ \end{matrix})) \end{matrix}$ $$\begin{array}{} \displaystyle (A+\widetilde A,B+\widetilde B)= \left( \left[ \begin{array}{c|cc} 1&\alpha&0\\ \varepsilon&\beta&0\\ \hline 0&0&1 \\ \end{array}\right], \left[ \begin{array}{cc|c} 0&1&0\\ \hline 0&0&1\\ \lambda&\mu&\delta \\ \end{array} \right] \right) \end{array}$$

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

$\begin{matrix} A \tilde{B} + \tilde{A} B = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ λ & μ & δ \end{matrix}) + (\begin{matrix} 0 & 0 & 0 \\ 0 & ε & β \\ 0 & 0 & 0 \end{matrix}) = (\begin{matrix} 0 & 0 & 0 \\ 0 & ε & β \\ λ & μ & δ \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle A\widetilde{B} +\widetilde{A}B= \begin{bmatrix} 0&0&0\\ 0&0&0\\ \lambda&\mu&\delta \end{bmatrix} \ + \ \ \begin{bmatrix} 0&0&0\\ 0&\varepsilon&\beta\\ 0&0&0 \end{bmatrix} \ = \ \begin{bmatrix} 0&0&0\\ 0&\varepsilon & \beta\\ \lambda&\mu&\delta \end{bmatrix}. \end{array}$$

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

$\begin{matrix} ((\begin{matrix} 1 & 0 & ε_{r} \\ ⋱ & ⋮ & 0 \\ 0 & 1 & ε_{2 r - 2} \\ ε_{1} & \dots & ε_{r - 1} & ε_{2 r - 1} \\ 0 & 1 & 0 \\ 0 & ⋱ & ⋱ \\ 0 & 0 & 1 \end{matrix}), (\begin{matrix} 0 & 1 & 0 \\ ⋱ & ⋱ & 0 \\ 0 & 0 & 1 \\ 1 & 0 \\ 0 & ⋱ \\ 0 & 1 \\ α_{1} & α_{2} & \dots & α_{r} & α_{r + 1} & \dots & α_{2 r - 1} \end{matrix})) \end{matrix}$ $$\begin{array}{} \displaystyle \left (\left[ \begin{array}{ccc|cccc} 1&&0&\varepsilon_{r}&\\ &\ddots&&\vdots&&0\\ 0&&1&\varepsilon_{2r-2}&\\ \varepsilon_1&\dots&\varepsilon_{r-1}&\varepsilon_{2r-1}&&&\\ \hline &&&0&1&&0\\ & \ \ 0&&&\ddots&\ddots&\\ &&&0&&0&1 \end{array}\right], \left[\begin{array}{cccc|ccc} 0&1&&0&&\\ &\ddots&\ddots&&&0\\ 0&&0&1&&\\ \hline &&&&1&&0\\ &&\!\!\!\!\!\!\!\!\!\!\!0&&&\ddots\\ &&&&0&&1\\ \alpha_1&\alpha_{2}&\dots&\alpha_{r}&\alpha_{r+1}&\dots& \alpha_{2r-1}\\ \end{array}\right] \right) \end{array}$$

in which all α_i and ε_j are independent parameters. Since the rth row of B is zero, a parameter ε_2r−2 does not appear in A͠B, and so in AB͠ + A͠B too. Choosing all parameters zeros except for ε_2r−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 (I_r, J_r(0)) ⊕ (J_r(0), I_r) for some r, or (F₁, G₁) ⊕ (G₂, F₂), or (F_m, G_m) ⊕ (G_m, F_m) for some m.

Since the deformation (4) is the direct sum of

$\begin{matrix} (I, ⨁_{i} (Φ (λ_{i}) + N)) and ((\begin{matrix} \oplus_{j} I_{r_{1 j}} & 0 & 0 \\ 0 & \oplus_{j} J_{r_{2 j}} (0) + N & N \\ 0 & N & \begin{matrix} P_{3} & N \\ 0 & Q_{4} \end{matrix} \end{matrix}), (\begin{matrix} \oplus_{j} J_{r_{1 j}} (0) + N & N & N \\ N & \oplus_{j} I_{r_{2 j}} & 0 \\ N & 0 & \begin{matrix} Q_{3} & 0 \\ N & P_{4} \end{matrix} \end{matrix})), \end{matrix}$ $$\begin{array}{} \displaystyle \Big(I,\bigoplus_i(\Phi({\lambda_i})+{N})\Big) \quad \text{and} \quad \left(\left[ \begin{array}{c|c|c} \oplus_j I_{r_{1j}}&0&0 \\\hline 0&\oplus_j J_{r_{2j}}(0)+N&N \\ \hline 0&N& \begin{matrix} P_3&N\\0&Q_4 \end{matrix} \end{array}\right], \left[\begin{array}{c|c|c} \oplus_j J_{r_{1j}}(0)+N &N & N\\ \hline N & \oplus_j I_{r_{2j}} &0\\ \hline N &0&\begin{matrix} Q_3&0\\ N&P_4 \end{matrix} \end{array}\right]\right), \end{array}$$

it is sufficient to consider (A, B) equals

$\begin{matrix} (I, ⨁_{i} (Φ (λ_{i}))) or ⨁_{j = 1}^{t_{1}} (I_{r_{1 j}}, J_{r_{1 j}}) \oplus ⨁_{j = 1}^{t_{2}} (J_{r_{2 j}}, I_{r_{2 j}}) \oplus ⨁_{j = 1}^{t_{3}} (F_{r_{3 j}}, G_{r_{3 j}}) \oplus ⨁_{j = 1}^{t_{4}} (G_{r_{4 j}}, F_{r_{4 j}}) . \end{matrix}$ $$\begin{array}{} \displaystyle \Big(I,\bigoplus_i(\Phi({\lambda_i}))\Big) \quad \text{or} \quad \bigoplus_{j=1}^{t_1} (I_{r_{1j}},J_{r_{1j}}) \oplus\bigoplus_{j=1}^{t_2}(J_{r_{2j}},I_{r_{2j}}) \oplus\bigoplus_{j=1}^{t_3}(F_{r_{3j}},G_{r_{3j}}) \oplus\bigoplus_{j=1}^{t_4}(G_{r_{4j}},F_{r_{4j}}). \end{array}$$(8)

Let first (A, B) = (I, ⨁_i(Φ(λ_i))). Then

$\begin{matrix} (A + \tilde{A}, B + \tilde{B}) = ((\begin{matrix} \oplus_{j} I_{r_{1 j}} & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & \oplus_{j} I_{r_{l j}} \end{matrix}),) ((\begin{matrix} \oplus_{j} J_{r_{1 j}} (λ_{1}) + N & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & \oplus_{j} J_{r_{l j}} (λ_{l}) + N \end{matrix})) . \end{matrix}$ $$\begin{array}{} \displaystyle (A+\widetilde{A},B+\widetilde{B})= \left (\left[ \begin{array}{c|c|c} \oplus_j I_{r_{1j}}&0&0\\ \hline 0&\ddots&0\\ \hline 0&0&\oplus_j I_{r_{lj}} \end{array}\right],\right . \left . \left[\begin{array}{c|c|c} \oplus_j J_{r_{1j}}(\lambda_1)+N&0&0\\ \hline 0&\ddots&0\\ \hline 0&0&\oplus_j J_{r_{lj}}(\lambda_l)+N \end{array}\right] \right). \end{array}$$

$\begin{matrix} \tilde{A} B + \tilde{A} B = \begin{matrix} (\begin{matrix} N & 0 & 0 \\ 0 & ⋱ & 0 \\ 0 & 0 & N \end{matrix}) \end{matrix} = 0, \end{matrix}$ $$\begin{array}{} \displaystyle \widetilde{A}B+\widetilde{A}B= \begin{array}{l} \left[\begin{array}{c|c|c} N&0&0\\ \hline 0&\ddots&0\\ \hline 0&0&N \end{array}\right] \end{array}=0, \end{array}$$

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

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

$\begin{matrix} P_{l} = {\bar{P}}_{l} + {\underline{P}}_{l}, Q_{l} = {\bar{Q}}_{l} + {\underline{Q}}_{l}, in which l = 3, 4, \end{matrix}$ $$\begin{array}{} \displaystyle P_l = \overline{P}_l+\underline{P}_l, \qquad Q_l = \overline{Q}_l+\underline{Q}_l, \quad \text{ in which } \, l=3,\,4, \end{array}$$

$\begin{matrix} {\bar{P}}_{l} = (\begin{matrix} F_{r_{l 1}} & 0 & \dots & 0 \\ F_{r_{l 2}} & ⋱ & ⋮ \\ ⋱ & 0 \\ 0 & F_{r_{l t_{l}}} \end{matrix}), {\underline{P}}_{l} = (\begin{matrix} H_{r_{l 1}} & H & \dots & H \\ H_{r_{l 2}} & ⋱ & ⋮ \\ ⋱ & H \\ 0 & H_{r_{l t_{l}}} \end{matrix}), \\ {\bar{Q}}_{l} = (\begin{matrix} G_{r_{l 1}} & 0 \\ 0 & G_{r_{l 2}} \\ ⋮ & ⋱ & ⋱ \\ 0 & \dots & 0 & G_{r_{l t_{l}}} \end{matrix}), {\underline{Q}}_{l} = (\begin{matrix} 0_{r_{l 1}} & 0 \\ H & 0_{r_{l 2}} \\ ⋮ & ⋱ & ⋱ \\ H & \dots & H & 0_{r_{l t_{l}}} \end{matrix}), \end{matrix}$ $$\begin{array}{} \displaystyle \overline{P}_l=\left[\!\!\!\begin{matrix} ~~F_{r_{l1}}&0&\cdots &0 \\ &F_{r_{l2}}& \ddots &\vdots\\ &&\ddots &0\\ {0}&&& F_{r_{lt_l}} \end{matrix}\!\!\!\right], \qquad\qquad\qquad\qquad\underline{P}_l=\left[\!\!\!\begin{matrix} ~~H_{r_{l1}}&H&\cdots &H \\ &H_{r_{l2}}& \ddots &\vdots\\ &&\ddots &H\\ {0}&&& H_{r_{lt_l}} \end{matrix}\!\!\!\right], \\ \displaystyle\overline{Q}_l=\left[\!\!\!\begin{matrix}~~G_{r_{l1}}&&& { 0} \\ 0& G_{r_{l2}}&& \\ \vdots&\ddots&\ddots&\\ 0&\cdots&0& G_{r_{lt_l}} \end{matrix}\!\!\!\right], \qquad\qquad\qquad\quad~~\,\underline{Q}_l=\left[\!\!\!\begin{matrix}~~ 0_{r_{l1}}&&& { 0} \\ H& 0_{r_{l2}}&& \\ \vdots&\ddots&\ddots&\\ H&\cdots&H& 0_{r_{lt_l}} \end{matrix}\!\!\!\right], \end{array}$$

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

Write

$\begin{matrix} J_{1} := \oplus_{j} J_{r_{1 j}} (0), J_{2} := \oplus_{j} J_{r_{2 j}} (0) . \end{matrix}$ $$\begin{array}{} \displaystyle J_1:=\oplus_j J_{r_{1j}}(0), \qquad J_2:=\oplus_j J_{r_{2j}}(0). \end{array}$$(9)

Then

$\begin{matrix} A = (\begin{matrix} I & 0 & 0 & 0 \\ 0 & J_{2} & 0 & 0 \\ 0 & 0 & {\bar{P}}_{3} & 0 \\ 0 & 0 & 0 & {\bar{Q}}_{4} \end{matrix}), & \tilde{A} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & N & N & N \\ 0 & N & {\underline{P}}_{3} & N \\ 0 & N & 0 & {\underline{Q}}_{4} \end{matrix}), \\ B = (\begin{matrix} J_{1} & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & {\bar{Q}}_{3} & 0 \\ 0 & 0 & 0 & {\bar{P}}_{4} \end{matrix}), & \tilde{B} = (\begin{matrix} N & N & N & N \\ N & 0 & 0 & 0 \\ N & 0 & {\underline{Q}}_{3} & 0 \\ N & 0 & N & {\underline{P}}_{4} \end{matrix}), \\ A \tilde{B} = (\begin{matrix} N & N & N & N \\ J_{2} N & 0 & 0 & 0 \\ {\bar{P}}_{3} N & 0 & {\bar{P}}_{3} {\underline{Q}}_{3} & 0 \\ {\bar{Q}}_{4} N & 0 & {\bar{Q}}_{4} N & {\bar{Q}}_{4} {\underline{P}}_{4} \end{matrix}), & \tilde{A} B = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & N & N {\bar{Q}}_{3} & N {\bar{P}}_{4} \\ 0 & N & {\underline{P}}_{3} {\bar{Q}}_{3} & N {\bar{P}}_{4} \\ 0 & N & 0 & {\underline{Q}}_{4} {\bar{P}}_{4} \end{matrix}), \end{matrix}$ $$\begin{array}{} \displaystyle {A}= \left[ \begin{array}{c|c|cc} I&0&0&0\\ \hline 0&J_2&0&0\\ \hline 0&0&\overline{P}_3 &0 \\ 0&0&0&\overline{Q}_4 \\ \end{array}\right], &&\qquad\qquad\widetilde{A}= \left[ \begin{array}{c|c|cc} 0&0&0&0\\ \hline 0&{N}&{N}&{N}\\ \hline 0&{N}&\underline{P}_3&{N}\\ 0&{N}&0&\underline{Q}_4 \end{array}\right],\\\displaystyle B= \left[ \begin{array}{c|c|cc} J_1&0&0&0\\ \hline 0&I&0&0\\ \hline 0&0&\overline{Q}_3 &0 \\ 0&0&0&\overline{P}_4 \\ \end{array}\right], &&\qquad\qquad\widetilde{B}= \left[ \begin{array}{c|c|cc} {N}&{N}&{N}&{N}\\ \hline {N}&0&0&0\\ \hline {N}&0&\underline{Q}_3&0\\ {N}&0&{N}&\underline{P}_4 \end{array}\right], \\\displaystyle A\widetilde{B}= \left[ \begin{array}{c|c|cc} {N}&{N}&{N}&{N}\\ \hline J_2{N}&0&0&0\\ \hline \overline{P}_3{N}&0& \overline{P}_3\underline{Q}_3&0 \\ \overline{Q}_4{N}&0& \overline{Q}_4{N}&\overline{Q}_4\underline{P}_4 \\ \end{array}\right], &&\qquad\qquad \widetilde{A}B= \left[ \begin{array}{c|c|cc} 0&0&0&0\\ \hline 0&{N}&{N}\overline{Q}_3&{N}\overline{P}_4\\ \hline 0&{N}&\underline{P}_3\overline{Q}_3&{N}\overline{P}_4\\ 0&{N}&0&\underline{Q}_4 \overline{P}_4 \\ \end{array}\right], \end{array}$$

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

$\begin{matrix} J_{2} N, N {\bar{P}}_{4}, {\bar{P}}_{3} N, N {\bar{Q}}_{3}, {\bar{Q}}_{4} N \end{matrix}$ $$\begin{array}{} \displaystyle J_2{N}, \quad {N}\overline{P}_4, \quad \overline{P}_3{N}, \quad N\overline{Q}_3, \quad \overline{Q}_4{N} \end{array}$$(10)

are nonzero if the corresponding parameter blocks N are nonzero.

Let us consider the first matrix in (10):

$\begin{matrix} J_{2} N = (\begin{matrix} J_{r_{1}} & 0 \\ J_{r_{2}} \\ ⋱ \\ 0 & J_{r_{n}} \end{matrix}) (\begin{matrix} H_{r_{1}} & 0 \\ H_{r_{2}} \\ ⋱ \\ 0 & H_{r_{n}} \end{matrix}) = (\begin{matrix} 0 \\ ε_{11} & \dots & ε_{1 m_{1}} \\ 0 & \dots & 0 \end{matrix}) \oplus \dots \oplus (\begin{matrix} 0 \\ ε_{n 1} & \dots & ε_{n m_{n}} \\ 0 & \dots & 0 \end{matrix}), \end{matrix}$ $$\begin{array}{} \displaystyle J_2N= \left[\begin{array}{cccc} J_{{r_1}}&& &0 \\ &J_{{r_2}}& &\\ &&\ddots&\\ {0}&&&J_{{{r_n}}} \end{array}\right] \left[\begin{array}{cccc} H_{r_{1}}&&&0\\ & H_{r_{2}}&&\\ &&\ddots &\\ {0}&&& H_{r_{{n}}} \end{array}\right] = \left[\begin{array}{ccc} &&\\ &\text{0}&\\ \varepsilon_{{11}}&\dots&\varepsilon_{{1{m_1}}}\\ 0&\dots&0\\ \end{array}\right] \oplus \dots \oplus \left[\begin{array}{ccc} &&\\ &\text{0}&\\ \varepsilon_{{n1}}&\dots&\varepsilon_{{n{m_n}}}\\ 0&\dots&0\\ \end{array}\right], \end{array}$$

in which all ε_ij are independent parameters and r₁ ⩽ r₂ ⩽ … ⩽ r_n. Clearly, J₂N ≠ 0 if at least one ε_ij ≠ 0.

Let us consider the second matrix in (10):

$\begin{matrix} N {\bar{P}}_{4} = (\begin{matrix} H_{r_{1}} & 0 \\ H_{r_{2}} \\ ⋱ \\ 0 & H_{r_{n}} \end{matrix}) (\begin{matrix} F_{r_{1}} & 0 \\ F_{r_{2}} \\ ⋱ \\ 0 & F_{r_{n}} \end{matrix}) = (\begin{matrix} 0 \\ ε_{11} & \dots & ε_{1 m_{1}} \end{matrix}) \oplus \dots \oplus (\begin{matrix} 0 \\ ε_{n 1} & \dots & ε_{n m_{n}} \end{matrix}) . \end{matrix}$ $$\begin{array}{} \displaystyle N\overline{P}_4= \left[\begin{array}{cccc} H_{{r_1}}&& &0 \\ &H_{{r_2}}& &\\ &&\ddots&\\ {0}&&&H_{{{r_n}}} \end{array}\right] \left[\begin{array}{cccc} F_{r_{1}}&&&0\\ & F_{r_{2}}&&\\ &&\ddots &\\ {0}&&& F_{r_{{n}}} \end{array}\right] = \left[\begin{array}{ccc} &&\\ &\text{0}&\\ \varepsilon_{{11}}&\dots&\varepsilon_{{1m_1}}\\ \end{array}\right] \oplus \dots \oplus \left[\begin{array}{ccc} &&\\ &\text{0}&\\ \varepsilon_{{n1}}&\dots&\varepsilon_{{n{m_n}}}\\ \end{array}\right]. \end{array}$$

in which all ε_j are independent parameters and r₁ ⩾ r₂ ⩾ … ⩾ r_n. Clearly, NP₄ ≠ 0 if at least one ε_ij ≠ 0.

The matrices P₃N, Q₄N, NQ₃, and Q₄N in (10) are considered analogously.

eISSN:: 2444-8656
Sprache:: Englisch

Zeitrahmen der Veröffentlichung:: Volume Open
Fachgebiete der Zeitschrift:: Biologie, andere, Mathematik, Angewandte Mathematik, Allgemeines, Physik

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Perturbation analysis of a matrix differential equation ẋ = ABx

Online veröffentlicht: 03. Okt. 2018

Seitenbereich: 97 - 104

Eingereicht: 09. Nov. 2017

Akzeptiert: 02. Apr. 2018

DOI: https://doi.org/10.21042/AMNS.2018.1.00007

SchlüsselwörterContragredient equivalence, Miniversal deformation, Perturbation

© 2018 M. Isabel García-Planas and Tetiana Klymchuk, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Schlüsselwörter
Contragredient equivalence, Miniversal deformation, Perturbation