Flows of matrix pencils^{∗}

Juan-Miguel Gracia^{†}

April 21, 1999

Abstract

We consider some matrix ordinary differential equations whose solutions preserve some features of the initial conditions; for example the Kronecker canonical form of a matrix pencil, or the Brunovsky canonical form of a matrix pair.

1 Introduction

It is well known that the existence and properties of the constant solutions x(t) ≡ c of an ordinary differential equation

ẋ(t) = f(t, x(t))  (1.1)

are a very important tool for the study of all solutions of this equation. Here t runs over an interval I of the real line ℝ.

The following more important solutions of (1.1) are the almost-constant solutions. Roughly we call so the solutions that keep constant something. For example, the periodic solutions keep constant the shape of the graphic on an periodic interval. Another example is the following: Suppose that t_{0} ∈ I and the decimal expansion of the real number x(t_{0}) is

x(t_{0}) = p.a_{1}a_{2} ⋯ a_{k}a_{k+1} ⋯

If for all t ≥ t_{0},

x(t) = p.a_{1}a_{2} ⋯ a_{k} ⋆ ⋯,

the solution x keeps constant the integral part and the first k decimal digits.

∗Supported by the Ministerio de Educación y Cultura (DGICYT), Proyecto PB97– 0599–CO3–01.

†Department of Applied Mathematics and Statistics, The University of the Basque Country, Faculty of Pharmacy, P.O. Box 450, E–01080 Vitoria-Gasteiz, Spain.

E-mail: mepgrmej@vc.ehu.es

When we consider a vector ordinary differential equation

˙x⃗(t) = f⃗(t, x⃗(t))

we can interest us in the solutions x⃗ that keep constant the direction

\frac{x⃗(t)}{∥x⃗(t)∥} = \frac{x⃗(t_{0})}{∥x⃗(t_{0})∥} for all t ≥ t_{0},

or the norm

∥x⃗(t)∥ = ∥x⃗(t_{0})∥ for all t ≥ t_{0}.

A special case of vector equations is that of matrix ordinary differential equations

Ẋ(t) = F (t, X(t)),

where the unknown X is a matrix function of a real variable t. This X(t) is a richer object than a vector. In this case we can study when there exist solutions that keep constant another features of the matrix X(t) as the rank rank X(t), or the determinant det X(t), or the trace tr X(t), or the spectrum σ(X(t)), or the singular values of X(t), or the invariant factors of λI − X(t) (when X(t) is a squared matrix), etc.

With respect to these questions we can consider the following problems: Problem 1. Characterization of the matrix C in order to the solution

of the initial value problem

\{ \begin{matrix}Ẋ(t) = F (t, X(t)), \\ X(t_{0}) = C, \end{matrix}

keep constant the rank, i.e.

rank X(t) = rank C for all t ≥ t_{0}.

There is a preliminary problem of existence: It can happen that no solution of the differential equation Ẋ(t) = F (t, X(t)) keep the rank.

Problem 2. Characterization of the matrix differential equations

Ẋ = F (t, X),

such that all its solutions keep the rank.

There are analogous problems changing the rank for, determinant, trace, spectrum (isospectral flows), etc. A general answer to Problem 2 for autonomous equations Ẋ = F (X) has relation with first integrals, obtained from the algebraic characterizations of rank, determinant, trace, characteristic polynomial, etc.

In this paper we give some results about the Problem 2 for X(t) keep constant:

• The Jordan canonical form of X(t), (Section 2).

• The Kronecker canonical form of the pencil λX(t) − Y (t), (Section 3).

• The Brunovsky canonical form of the matrix pair (A(t), B(t)), (Section 4).

2 Self-similar flows of matrices

Theorem 2.1 Let Λ : I → ℂ^{n×n} be a matrix function defined on an interval I of ℝ. Let t_{0} be a point of I. Let X : I → ℂ^{n×n} be any solution of the matrix differential equation

Ẋ(t) = Λ(t) X(t) − X(t) Λ(t).  (2.1)

Then for all t ∈ I, the matrix X(t) is similar to the matrix X(t_{0}).

Theorem 2.2 Let Z : I → ℂ^{n×n} be a matrix function of class C^{1} defined on an interval I of ℝ. Let t_{0} ∈ I. Suppose that for all t ∈ I, the matrix Z(t) is similar to the matrix Z(t_{0}). Then there exists a continuous matrix function Γ : I → ℂ^{n×n} such that for all t ∈ I

Ż(t) = Γ(t) Z(t) − Z(t) Γ(t).  (2.2)

Proof of Theorem 2.1 Let T (t) be the unique solution of the initial value problem

\{ \begin{matrix}Ṫ (t) & = & Λ(t) T (t), \\ T (t_{0}) & = & I_{n}. \end{matrix}  (2.3)

As T (t) is a fundamental matrix at t_{0}, it follows that det T (t) ≠ 0 for all t ∈ I, [4, p. 152, Teor. 2.5].

Call Y (t) := T (t) X(t_{0}) T (t)^{−1}, for all t ∈ I.

Then deriving we have that for all t ∈ I,

Ẏ (t) = Ṫ (t) X(t_{0}) T (t)^{−1} − T (t) X(t_{0}) T (t)^{−1} Ṫ (t) T (t)^{−1}
= Ṫ (t) T (t)^{−1} T (t) X(t_{0}) T (t)^{−1} − T (t) X(t_{0}) T (t)^{−1} Ṫ (t) T (t)^{−1};

from here by (2.3), we deduce

Ẏ (t) = Λ(t) Y (t) − Y (t) Λ(t),  (2.4)

for all t ∈ I.

In addition

Y (t_{0}) = T (t_{0}) X(t_{0}) T (t_{0})^{−1} = X(t_{0}).

By the uniqueness theorem of solutions of differential equation (2.1) we have

X(t) = Y (t),

for all t ∈ I.

Therefore, for all t ∈ I,

X(t) = T (t) X(t_{0}) T (t)^{−1}.
□

Proof of Theorem 2.2

Given that for all t ∈ I the matrix Z(t) is similar to the matrix Z(t_{0}) , the matrix Z(t) has a constant Segre characteristic. Therefore, there exists a matrix function of class C^{1}, P : I → ℂ^{n×n} such that for all t ∈ I:

1. the matrix P (t) is invertible, and

2. P (t)^{−1} Z(t) P (t) = J(t) (Jordan form). See for ex. [2, p. 374, Corol. 3.9].

But J(t) = J(t_{0}) according to Z(t) ≈ Z(t_{0}), (≈ means is similar to). Hence for all t ∈ I,

Z(t) = P (t) J(t_{0}) P (t)^{−1} = P (t) P (t_{0})^{−1} Z(t_{0}) P (t_{0}) P (t)^{−1};

what implies for all t ∈ I,

Z(t) = Q(t) Z(t_{0}) Q(t)^{−1}  (2.5)

where

Q(t) := P (t) P (t_{0})^{−1}.

From (2.5) we deduce deriving that

Ż(t) = Q̇(t) Z(t_{0}) Q(t)^{−1} − Q(t) Z(t_{0}) Q(t)^{−1} Q̇(t) Q(t)^{−1}
= Q̇(t) Q(t)^{−1} Q(t) Z(t_{0}) Q(t)^{−1} − Q(t) Z(t_{0}) Q(t)^{−1} Q̇(t) T (t)^{−1}.

Hence,

Ż(t) = (Q̇(t) Q̇(t)^{−1}) Z(t) − Z(t) (Q̇(t) Q̇(t)^{−1}).

Calling,

Γ(t) := Q̇(t) Q̇(t)^{−1},

we have that for all t ∈ I

Ż(t) = Γ(t) Z(t) − Z(t) Γ(t).
□

3 Flows of strictly equivalent pencils

Recall that two matrix pencils λF_{1}−G_{1} and λF_{2}−G_{2} , with F_{1}, G_{1}, F_{2}, G_{2} ∈ ℂ^{m×n}, are called strictly equivalent if there exist invertible matrices P ∈ ℂ^{m×m} and Q ∈ ℂ^{n×n} such that

λF_{2} − G_{2} = P (λF_{1} − G_{1})Q.

Theorem 3.1 Let I be an interval of ℝ, and t_{0} ∈ I. Let U : I → ℂ^{m×m} and V : I → ℂ^{n×n} be continuous matrix functions. Let X, Y be two solutions of equation

Ż(t) = U(t) Z(t) + Z(t) V (t)  (3.1)

Then for all t ∈ I the pencil λX(t) − Y (t) is strictly equivalent to the pencil λX(t_{0}) − Y (t_{0}).

Proof. Define

P (t) the solution of \{ \begin{matrix}Ṗ (t) & = & U(t) P (t) \\ P (t_{0}) & = & I_{m} \end{matrix}

Q(t) the solucion of \{ \begin{matrix}Q̇(t) & = & Q(t) V (t) \\ Q(t_{0}) & = & I_{n} \end{matrix}

This implies that P (t) and Q(t) are of class C^{1} and invertible for all t ∈ I. Now define

\begin{aligned}X_{1}(t) & := & P (t) X(t_{0}) Q(t) \\ Y_{1}(t) & := & P (t) Y (t_{0}) Q(t) \end{aligned}

what implies λX_{1}(t) − Y_{1}(t) = P (t) (λX(t_{0}) − Y (t_{0})) Q(t) for all t ∈ I. In addition,

Ẋ_{1}(t) = Ṗ (t) P^{−1}(t) P (t) X(t_{0}) Q(t) + P (t) X(t_{0}) Q(t) Q^{−1}(t) Q̇(t)
= U(t) X_{1}(t) + X_{1}(t) V (t),

what implies X_{1}(t) is solution of (3.1).

Given that

X_{1}(t_{0}) = P (t_{0}) X(t_{0}) Q(t_{0}) = I_{m} X(t_{0}) I_{n} = X(t_{0}),

by the uniqueness of the solutions

X_{1}(t) = X(t)

for all t ∈ I.

Analogously,

Ẏ_{1}(t) = U(t) Y_{1}(t) + Y_{1}(t) V (t)
Ẏ_{1}(t_{0}) = P (t_{0}) Y (t_{0}) Q(t_{0}) = I_{m} Y (t_{0}) I_{n} = Y (t_{0})

that, by the uniqueness of the solutions, implies

Y_{1}(t) = Y (t) for all t ∈ I.

From that,

λX(t) − Y (t) = P (t) (λX(t_{0}) − Y (t_{0})) Q(t)

for all t ∈ I.

□

Moreover Lemma 1.22 p. 139 of [3] implies that for all t ∈ I rank X(t) = rank X(t_{0}) and rank Y (t) = rank Y (t_{0}).

Theorem 3.2 Let F, G : I → ℂ^{m×n} be matrix functions of class C^{1} defined on an interval I ⊂ ℝ. Let t_{0} ∈ I. Suppose that for all t ∈ I, the pencil λF (t) − G(t) is strictly equivalent to the pencil λF (t_{0}) − G(t_{0}). Then there exist continuous matrix functions U : I → ℂ^{m×m}, V : I → ℂ^{n×n} such that for all t ∈ I,

Ḟ (t) = U(t) F (t) + F (t) V (t)
Ġ(t) = U(t) G(t) + G(t) V (t).

Proof.

By the hypothesis, the Kronecker structure of the pencil λF (t) − G(t) is constant on I. From that, there exist matrix functions P : I → ℂ^{m×m}, Q : I → ℂ^{n×n} of class C^{1} such that for all t ∈ I, P (t) and Q(t) are invertible matrices and

P (t) (λF (t) − G(t)) Q(t) = λF_{c}(t) − G_{c}(t)

(where λF_{c}(t) − G_{c}(t) is a pencil in Kronecker canonical form ). As for all t ∈ I,

λF_{c}(t) − G_{c}(t) = λF_{c}(t_{0}) − G_{c}(t_{0}),

it follows that

λF (t) − G(t) = P (t)^{−1} (λF_{c}(t_{0}) − G_{c}(t_{0})) Q(t)^{−1}
= P (t)^{−1} P (t_{0})(λF (t_{0}) − G(t_{0})) Q(t_{0}) Q(t)^{−1}.

Calling

𝒫(t) := P (t)^{−1} P (t_{0}); 𝒬(t) := Q(t_{0}) Q(t)^{−1}

it results

λF (t) − G(t) = 𝒫(t) (λF (t_{0}) − G(t_{0})) 𝒬(t), t ∈ I.

By deriving with respect to t, it follows from here

λḞ (t) − Ġ(t) = 𝒫̇(t) (λF (t_{0}) − G(t_{0})) 𝒬(t) + 𝒫(t) (λF (t_{0}) − G(t_{0})) 𝒬(t)
= 𝒫̇(t) 𝒫(t)^{−1} 𝒫(t) (λF (t_{0}) − G(t_{0})) 𝒬(t)
+ 𝒫(t) (λF (t_{0}) − G(t_{0})) 𝒬(t) 𝒬(t)^{−1} 𝒬̇(t)
= (𝒫̇(t) 𝒫(t)^{−1}) (λF (t) − G(t))
+ (λF (t) − G(t)) (𝒬(t)^{−1} 𝒬̇(t)).

Calling

U(t) := 𝒫̇(t) 𝒫(t)^{−1} y V (t) := 𝒬(t)^{−1} 𝒬̇(t);

we have

Ḟ (t) = U(t) F (t) + F (t) V (t)
Ġ(t) = U(t) G(t) + G(t) V (t)

for all t ∈ I.

□

4 Self-blocksimilar flows

Recall that two matrix pairs (A_{1}, B_{1}), (A_{2}, B_{2}) ∈ ℂ^{n×n}×ℂ^{n×m} are blocksimilar (or feedback equivalent) if there exist invertible matrices P ∈ ℂ^{n×n}, Q ∈ ℂ^{m×m} and F ∈ ℂ^{m×n} such that

P [A_{2}, B_{2}] = [A_{1}, B_{1}] [ \begin{matrix}P & 0 \\ F & Q \end{matrix} ] .

We will write (A_{1}, B_{1}) ≈^{b} (A_{2}, B_{2}), where ≈^{b} means is blocksimilar to.

Theorem 4.1 Let U : I → ℂ^{n×n}, V : I → ℂ^{m×n}, W : I → ℂ^{m×m} be continuous matrix functions defined on an interval I of ℝ. Then every solution (A, B) of matrix differential equation

[Ȧ(t), Ḃ(t)] = U(t) [A(t), B(t)] + [A(t), B(t)] [\begin{matrix}−U(t) & 0 \\ V (t) & W (t) \end{matrix}]  (4.1)

is self-blocksimilar, i.e. given a t_{0} ∈ I one has

(A(t), B(t)) ≈^{b} (A(t_{0}), B(t_{0}))

for all t ∈ I.

Differential equation (4.1) is equivalent to the system of matrix differential equations

\{ \begin{matrix}Ȧ & = & U(t) A − A U(t) + B V (t) \\ Ḃ & = & U(t) B + B W (t). \end{matrix}  (4.2)

(Later we will try to find explicitly the solutions of (4.1) when U, V and W are constant).

Theorem 4.2 Let Ã : I → ℂ^{n×n}, B̃ : I → ℂ^{n×m} be matrix functions of class C^{1} defined on an interval I of ℝ. Let t_{0} ∈ I. Suppose that for all t ∈ I, the matrix pair (Ã(t), B̃(t)) is blocksimilar to the matrix pair (Ã(t_{0}), B̃(t_{0})). Then there exist continuous matrix functions Ũ : I → ℂ^{n×n}, Ṽ : I → ℂ^{m×n} and W̃ : I → ℂ^{m×m} such that for all t ∈ I,

[ ˙Ã(t), ˙ B̃(t)] = Ũ(t) [Ã(t), B̃(t)] + [Ã(t), B̃(t)] [\begin{matrix}−Ũ(t) & 0 \\ Ṽ (t) & W̃ (t) \end{matrix}] .

Proof of Theorem 4.1

Let (A_{1}(t), B_{1}(t)), A_{1} : I → ℂ^{n×n}, B_{1} : I → ℂ^{n×m} a solution of differential equation (4.1). We will try to prove that for all t ∈ I we have

(A_{1}(t), B_{1}(t)) ≈^{b} (A_{1}(t_{0}), B_{1}(t_{0})).

Let P : I → ℂ^{n×n} be the solution of the initial value problem

\{ \begin{matrix}Ṗ (t) & = & U(t) P (t) \\ P (t_{0}) & = & I_{n}. \end{matrix}  (4.3)

Let Q : I → ℂ^{(n+m)×(n+m)} be the solution of the initial value problem

\{ \begin{matrix}Q̇(t) & = & Q(t) [\begin{matrix}−U(t) & 0 \\ V (t) & W (t) \end{matrix}] \\ Q(t_{0}) & = & I_{n+m}. \end{matrix}  (4.4)

Consider the matrix Q(t) partitioned as follows

Q(t) = \begin{matrix}n & m \\ [\begin{matrix}Q_{11}(t) & Q_{12}(t) \\ Q_{21}(t) & Q_{22}(t) \end{matrix}] & \begin{matrix}n \\ m \end{matrix} \end{matrix}

Then problem (4.4) is equivalent to problem:

\{ \begin{matrix}Q̇_{11}(t) & = & −Q_{11}(t) U(t) + Q_{12}(t) V (t), \\ Q_{11}(t_{0}) & = & I_{n} \end{matrix}  (4.5)

\{ \begin{matrix}Q̇_{12}(t) & = & Q_{12}(t) W (t), \\ Q_{12}(t_{0}) & = & 0 \end{matrix}  (4.6)

\{ \begin{matrix}Q̇_{21}(t) & = & −Q_{21}(t) U(t) + Q_{22}(t) V (t), \\ Q_{21}(t_{0}) & = & 0 \end{matrix}  (4.7)

\{ \begin{matrix}Q̇_{22}(t) & = & Q_{22}(t) W (t), \\ Q_{22}(t_{0}) & = & I_{m}. \end{matrix}  (4.8)

From (4.6) by the uniqueness of the solutions it results Q_{12}(t) ≡ 0. Hence

Q(t) = [\begin{matrix}Q_{11}(t) & 0 \\ Q_{21}(t) & Q_{22}(t) \end{matrix}]

for all t ∈ I.

In addition , as P (t_{0}) = I_{n} and Q(t_{0}) = I_{n+m}, it follows that P (t) and Q(t) are invertible for all t ∈ I. (See [4, p. 152, Teorema 2.5]). Moreover P, Q ∈ C^{1} on I.

Now define for all t ∈ I,

[A_{2}(t), B_{2}(t)] := P (t) [A_{1}(t_{0}), B_{1}(t_{0})] [\begin{matrix}P (t)^{−1} & 0 \\ Q_{21}(t) & Q_{22}(t) \end{matrix}] .

Hence

\{ \begin{matrix}A_{2}(t) & := & P (t) A_{1}(t_{0}) P (t)^{−1} + P (t) B_{1}(t_{0}) Q_{21}(t) \\ B_{2}(t) & := & P (t) B_{1}(t_{0}) Q_{22}(t). \end{matrix}  (4.9)

Deriving (4.9) with respect to t it results

Ȧ_{2}(t) = Ṗ (t) A_{1}(t_{0}) P (t)^{−1} − P (t) A_{1}(t_{0}) P^{−1}(t) Ṗ (t) P^{−1}(t)
+Ṗ (t) B_{1}(t_{0}) Q_{21}(t) + P (t) B_{1}(t_{0}) Q̇_{21}(t),  (4.10)
Ḃ_{2}(t) = Ṗ (t) B_{1}(t_{0}) Q_{22}(t) + P (t) B_{1}(t_{0}) Q̇_{22}(t).  (4.11)

From (4.3), we deduce

Ṗ (t) P (t)^{−1} = U(t)

and from (4.7)

Q̇_{21}(t) = −Q_{21}(t) U(t) + Q_{22}(t) V (t).

From here and (4.10) we deduce

Ȧ_{2}(t) = Ṗ (t) P (t)^{−1} P (t) A_{1}(t_{0}) P (t)^{−1} − P (t) A_{1}(t_{0}) P (t)^{−1} Ṗ (t) P (t)^{−1}
+ Ṗ (t) P (t)^{−1} P (t) B_{1}(t_{0}) Q_{21}(t)
+ P (t) B_{1}(t_{0}) [−Q_{21}(t) U(t) + Q_{22}(t) V (t)].

Hence,

Ȧ_{2}(t) = U(t) [P (t) A_{1}(t_{0}) P (t)^{−1} + P (t) B_{1}(t_{0}) Q_{21}(t)]
− [P (t) A_{1}(t_{0}) P (t)^{−1} + P (t) B_{1}(t_{0}) Q_{21}(t)] U(t)
+ P (t) B_{1}(t_{0}) Q_{22}(t) V (t)
= U(t) A_{2}(t) − A_{2}(t) U(t) + B_{2}(t) V (t).  (4.12)

From (4.3), (4.8) and (4.11),it follows

Ḃ_{2}(t) = Ṗ (t) P (t)^{−1} P (t) B_{1}(t_{0}) Q_{22}(t) + P (t) B_{1}(t_{0}) Q_{22}(t) W (t)
= U(t) B_{2}(t) + B_{2}(t) W (t).  (4.13)

Therefore (A_{2}(t), B_{2}(t)) is a solution of differential equation (4.1). Furthermore,

[A_{2}(t_{0}), B_{2}(t_{0})] = I_{n} [A_{1}(t_{0}), B_{1}(t_{0})] [\begin{matrix}I_{n} & 0 \\ 0 & I_{m} \end{matrix} ] = [A_{1}(t_{0}), B_{1}(t_{0})].

It results from here —by the uniqueness of the solutions— that for all t ∈ I,

[A_{2}(t), B_{2}(t)] = [A_{1}(t), B_{1}(t)].

What, by the definition of A_{2} and B_{2}, implies that for all t ∈ I,

[A_{1}(t), B_{1}(t)] = P (t) [A_{1}(t_{0}), B_{1}(t_{0})] [\begin{matrix}P (t)^{−1} & 0 \\ Q_{21}(t) & Q_{22}(t) \end{matrix}] .

Therefore, for all t ∈ I, the pair (A_{1}(t), B_{1}(t)) is blocksimilar to the pair (A_{1}(t_{0}), B_{1}(t_{0})).

□

Proof of Theorem 4.2

For every t ∈ I, the pair (Ã(t), B̃(t)) is blocksimilar to (Ã(t_{0}), B̃(t_{0})). In consequence, the controllability indices and the Segre characteristic of (Ã, B̃) are constant. Hence there exist matrix functions of class C^{1}, P_{1} : I → ℂ^{n×n}, Q_{1} : I → ℂ^{m×m} and R_{1} : I → ℂ^{m×n} such that for all t ∈ I:

P_{1}(t) y Q_{1}(t)  (4.14)

are invertible matrices and

P_{1}(t)^{−1} [Ã(t), B̃(t)] [\begin{matrix}P_{1}(t) & 0 \\ R_{1}(t) & Q_{1}(t) \end{matrix}] = [A_{c}(t), B_{c}(t)]  (4.15)

where (A_{c}(t), B_{c}(t)) is a pair in the Brunovsky canonical form.

As

[A_{c}(t), B_{c}(t)] = [A_{c}(t_{0}), B_{c}(t_{0})]

for all t ∈ I because

(Ã(t), B̃(t)) ≈^{b} (Ã(t_{0}), B̃(t_{0}))

for all t ∈ I, it follows that for all t ∈ I,

[Ã(t), B̃(t)] = P_{1}(t) [A_{c}(t_{0}), B_{c}(t_{0})] [\begin{matrix}P_{1}(t) & 0 \\ R_{1}(t) & Q_{1}(t) \end{matrix}]^{−1}
= P_{1}(t) P_{1}(t_{0})^{−1} [Ã(t_{0}), B̃(t_{0})]
[\begin{matrix}P_{1}(t_{0}) & 0 \\ R_{1}(t_{0}) & Q_{1}(t_{0}) \end{matrix}] [\begin{matrix}P_{1}(t) & 0 \\ R_{1}(t) & Q_{1}(t) \end{matrix}]^{−1} ;  (4.16)

Let us interest in the inverse of the matrix

[\begin{matrix}P_{1}(t) & 0 \\ R_{1}(t) & Q_{1}(t) \end{matrix}]

as a preliminary question: Let be the matrix

M = [\begin{matrix}M_{11} & 0 \\ M_{21} & M_{22} \end{matrix} ] ∈ ℂ^{(n+m)×(n+m)}

with M_{11} ∈ ℂ^{n×n} and M_{22} ∈ ℂ^{m×m} invertible matrices. Let

X = [\begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ]

be the inverse matrix of M, where we consider that X is partitioned as M (i.e. X_{11} ∈ ℂ^{n×n}, X_{22} ∈ ℂ^{m×m}).

From MX = I_{n+m} = XM we deduce

[\begin{matrix}M_{11} & 0 \\ M_{21} & M_{22} \end{matrix} ] [\begin{matrix}X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix} ] = [ \begin{matrix}M_{11}X_{11} & M_{11}X_{12} \\ M_{21}X_{11} + M_{22}X_{21} & M_{21}X_{12} + M_{22}X_{22} \end{matrix} ]
= [ \begin{matrix}I_{n} & 0 \\ 0 & I_{m} \end{matrix} ]

⟹ M_{11}X_{12} = 0 ⟹ M_{11}^{−1}M_{11}X_{12} = 0 ⟹ X_{12} = 0 ⟹ M_{22}X_{22} = I_{m}

[\begin{matrix}X_{11} & 0 \\ X_{21} & X_{22} \end{matrix} ] [\begin{matrix}M_{11} & 0 \\ M_{21} & M_{22} \end{matrix} ] = [ \begin{matrix}X_{11}M_{11} & 0 \\ X_{21}M_{11} + X_{22}M_{21} & X_{22}M_{22} \end{matrix} ]
= [ \begin{matrix}I_{n} & 0 \\ 0 & I_{m} \end{matrix} ]

⟹ M_{11}X_{11} = I_{n} ⟹ X_{11} = M_{11}^{−1}; X_{22}M_{22} = I_{m} ⟹ X_{22} = M_{22}^{−1} M_{21}X_{11} + M_{22}X_{21} = 0 ⟹ M_{21}M_{11}^{−1} = −M_{22}X_{21} ⟹ M_{22}^{−1}M_{21}M_{11}^{−1} = X_{21} X_{21}M_{11} = −M_{22}^{−1}M_{21} ⟹ X_{21} = −M_{22}^{−1}M_{21}M_{11}^{−1} ⟹

[\begin{matrix}M_{11} & 0 \\ M_{21} & M_{22} \end{matrix} ]^{−1} = [ \begin{matrix}M_{11}^{−1} & 0 \\ M_{22}^{−1}M_{21}M_{11}^{−1} & M_{22}^{−1} \end{matrix} ]

Thus, from (4.16) we deduce

[Ã(t), B̃(t)] = P_{1}(t) P_{1}(t_{0})^{−1} [Ã(t_{0}), B̃(t_{0})]
[\begin{matrix}P_{1}(t_{0}) & 0 \\ R_{1}(t_{0}) & Q_{1}(t_{0}) \end{matrix}] [ \begin{matrix}P_{1}(t)^{−1} & 0 \\ Q_{1}(t)^{−1} R_{1}(t) P_{1}(t)^{−1} & Q_{1}(t)^{−1} \end{matrix} ]
= P_{1}(t) P_{1}(t_{0})^{−1} [Ã(t_{0}), B̃(t_{0})]

[ \begin{matrix}P_{1}(t_{0}) P_{1}(t)^{−1} & 0 \\ R_{1}(t_{0}) P_{1}(t)^{−1} + Q_{1}(t_{0}) Q_{1}(t)^{−1} R_{1}(t) P_{1}(t)^{−1} & Q_{1}(t_{0}) Q_{1}(t)^{−1} \end{matrix} ] .

Calling for all t ∈ I,

𝒫(t) := P_{1}(t) P_{1}(t_{0})^{−1},  𝒬(t) := Q_{1}(t_{0}) Q_{1}(t)^{−1}
ℛ(t) := R_{1}(t_{0}) P_{1}(t)^{−1} + Q_{1}(t_{0}) Q_{1}(t)^{−1} R_{1}(t) P_{1}(t)^{−1}
:= R_{1}(t_{0}) P_{1}(t)^{−1} + 𝒬(t) R_{1}(t) P_{1}(t)^{−1},

⟹ 𝒫, 𝒬, ℛ ∈ C^{1} it results that for all t ∈ I,

[Ã(t), B̃(t)] = 𝒫(t) [Ã(t_{0}), B̃(t_{0})] [\begin{matrix}𝒫(t)^{−1} & 0 \\ ℛ(t) & 𝒬(t) \end{matrix}]

What is equivalent to

Ã(t) = 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} + 𝒫(t) B̃(t_{0}) ℛ(t)  (4.17)
B̃(t) = 𝒫(t) B̃(t_{0}) 𝒬(t).  (4.18)

Deriving (4.17) with respect to t, we deduce

˙Ã(t) = 𝒫̇(t) 𝒫^{−1}(t) 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} − 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} 𝒫̇(t) 𝒫(t)^{−1}
+ 𝒫̇(t) 𝒫^{−1}(t) 𝒫(t) B̃(t_{0}) ℛ(t) + 𝒫(t) B̃(t_{0}) ℛ̇(t)
= 𝒫̇(t) 𝒫(t)^{−1} [𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} + 𝒫(t) B̃(t_{0}) ℛ(t)]
− 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} 𝒫̇(t) 𝒫(t)^{−1} + 𝒫(t) B̃(t_{0}) ℛ̇(t)
= 𝒫̇(t) 𝒫(t)^{−1} Ã(t) − 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} 𝒫̇(t) 𝒫(t)^{−1}
+ 𝒫(t) B̃(t_{0}) ℛ̇(t).  (4.19)

In (4.19) we have made use of

Ṙ(t) = − R_{1}(t_{0}) P_{1}(t)^{−1} Ṗ_{1}(t) P_{1}(t)^{−1} − 𝒬(t) R_{1}(t) P_{1}(t)^{−1} Ṗ_{1}(t) P_{1}(t)^{−1}
+ 𝒬̇(t) R_{1}(t) P_{1}(t)^{−1} + 𝒬(t) Ṙ_{1}(t) P_{1}(t)^{−1}
= − ℛ(t) Ṗ_{1}(t) P_{1}(t)^{−1} + 𝒬̇(t) R_{1}(t) P_{1}(t)^{−1}
+ 𝒬(t) Ṙ_{1}(t) P_{1}(t)^{−1}.  (4.20)

But

𝒬̇(t) = −Q_{1}(t_{0}) Q_{1}(t)^{−1} Q̇_{1}(t) Q_{1}(t)^{−1} = −𝒬(t) Q̇_{1}(t) Q_{1}(t)^{−1}; hence —according to (4.20)—

ℛ̇(t) = −ℛ(t) Ṗ_{1}(t) P_{1}(t)^{−1} + 𝒬(t) [−Q̇_{1}(t) Q_{1}(t)^{−1} + Ṙ_{1}(t) P_{1}(t)^{−1}]. Returning (4.19) we obtain

˙Ã(t) = 𝒫̇(t) 𝒫(t)^{−1} Ã(t) − 𝒫(t) Ã(t_{0}) 𝒫(t)^{−1} 𝒫̇(t) 𝒫(t)^{−1}
− 𝒫(t) B̃(t_{0}) ℛ(t) Ṗ_{1}(t) P_{1}(t)^{−1} + 𝒫(t) B̃(t_{0}) 𝒬(t) Ṽ (t), (4.21)

having called

Ṽ (t) := −Q̇_{1}(t) Q_{1}(t)^{−1} + Ṙ_{1}(t) P_{1}(t)^{−1}.

As

𝒫̇(t) 𝒫(t)^{−1} = Ṗ_{1}(t) P_{1}(t_{0})^{−1} P_{1}(t_{0}) P_{1}(t)^{−1} = Ṗ_{1}(t) P_{1}(t)^{−1},

the identities (4.17), (4.18) y (4.21) imply

˙Ã(t) = 𝒫̇(t) 𝒫(t)^{−1} Ã(t) − Ã(t) 𝒫̇(t) 𝒫(t)^{−1} + B̃(t) Ṽ (t);  (4.22)

and calling

Ũ(t) := 𝒫̇(t) 𝒫(t)^{−1}

(4.22) changes to

˙Ã(t) = Ũ(t) Ã(t) − Ã(t) + B̃(t) Ṽ (t).  (4.23)

Deriving (4.18) with respect to t, one obtain

˙B̃(t) = 𝒫̇(t) 𝒫(t)^{−1} 𝒫(t) B̃(t_{0}) 𝒬(t)
+ 𝒫(t) B̃(t_{0}) [−𝒬(t) Q̇_{1}(t) Q_{1}(t)^{−1}]

= Ũ(t) B̃(t) + B̃(t) W̃ (t)  (4.24)

where

W̃ (t) := −Q̇_{1}(t) Q_{1}(t)^{−1}.

Therefore,

[ ˙Ã(t), ˙ B̃(t)] = Ũ(t) [Ã(t), B̃(t)] + [Ã(t), B̃(t)] [\begin{matrix}−Ũ(t) & 0 \\ Ṽ (t) & W̃ (t) \end{matrix}]

□

Solution when the coefficients of the equation

[Ȧ(t), Ḃ(t)] = U(t) [A(t), B(t)] + [A(t), B(t)] [\begin{matrix}−U(t) & 0 \\ V (t) & W (t) \end{matrix}] (4.25)

are constant.

From now on let U ∈ ℂ^{n×n}, V ∈ ℂ^{m×n} and W ∈ ℂ^{m×m} be constant matrices. The linear matrix differential equation turns into

\{ \begin{matrix}Ȧ(t) & = & U A(t) − A(t) U + B(t) V \\ Ḃ(t) & = & U B(t) + B(t) W. \end{matrix}  (4.26)

We are going to try to obtain a formula that gives us the the solution of ( 4.26) that satisfies the initial conditions

\{ \begin{matrix}A(0) & = & A_{0} \\ B(0) & = & B_{0} \end{matrix}  (4.27)

where A_{0} ∈ ℂ^{n×n}, B_{0} ∈ ℂ^{n×m} are beforehand given matrices. We will proceed according the following manner. Firstly, we will write the solution of the initial value homogeneous problem

\{ \begin{matrix}Ḃ(t) & = & U B(t) + B(t) W \\ B(0) & = & B_{0}. \end{matrix}  (4.28)

This solution is

B(t) = e^{Ut} B_{0} e^{W t},

t ∈ ℝ, because

Ḃ(t) = U e^{Ut} B_{0} e^{W t} + e^{Ut} B_{0} e^{W t} W

y

B(0) = e^{U0} B_{0} e^{W 0} = I_{n} B_{0} I_{m} = B_{0}.

After we will solve the initial value nonhomogeneous problem

\{ \begin{matrix}Ȧ(t) & = & U A(t) − A(t) U + e^{Ut} B_{0} e^{W t} V \\ A(0) & = & A_{0}. \end{matrix}  (4.29)

[Parenthetical comment.

The solution of

\frac{dX}{dt} = AX + XB + F (t), X(0) = C

is given by

X(t) = e^{At} (C + ∫_{0}^{t} e^{−As} F (s) e^{−Bs} ds) e^{Bt}.

[1] ].

Hence, the solution of (4.29) will be

A(t) = e^{Ut} (A_{0} + ∫_{0}^{t} e^{−Us} e^{Us} B_{0} e^{W s} V e^{Us} ds) e^{−Ut}

= e^{Ut} (A_{0} + B_{0} ∫_{0}^{t} e^{W s} V e^{Us} ds) e^{−Ut},

what implies

(A(t), B(t)) = (e^{Ut} [A_{0} + B_{0} ∫_{0}^{t} e^{W s} V e^{Us} ds] e^{−Ut}, e^{Ut} B_{0} e^{W t})

for all t ∈ ℝ.

This make clear that (A(t), B(t)) ≈^{b} (A_{0}, B_{0}), because e^{Ut} and e^{W t} are invertible matrices for all t ∈ ℝ. Problem 3.

Let (A_{1}, B_{1}) ∈ ℂ^{n×n} × ℂ^{n×m} be a pair blocksimilar to (A_{0}, B_{0}). Are there U ∈ ℂ^{n×n}, V ∈ ℂ^{m×n} and W ∈ ℂ^{m×m} such that

A_{1} = e^{U} [A_{0} + B_{0} ∫_{0}^{1} e^{W s} V e^{Us} ds] e^{−U}
B_{1} = e^{U} B_{0} e^{W}?

An affirmative answer to this question let us to say that the equivalence classes for the blocksimilarity were arc-connected (even more, connected by C^{∞}–arcs). This answer is based on the concept of logarithm of a invertible matrix. But there is another proof of the connection of any blocksimilarity class, based on the fact that the linear general group GL_{q}(ℂ) is connected ([3, p. 352, Example 7.1]).

References

[1] R. Bellman, Linear Multilinear Algebra,13 (1983)73–79.

[2] J.C. Evard, J.M. Gracia, On similarities of class C^{p} and applications to matrix differential equations, Linear Algebra Appl.,137/138, (1990) 363–386.

[3] U. Helmke, J.B. Moore, Optimization and Dynamical Systems, Springer, 1994, 1996.

[4] O. Plaat, Ecuaciones Diferenciales Ordinarias, Reverté, Barcelona, 1974.