23.6 Matrix equations

To solve the following exercises you need to know the elementary properties of the exponential of matrices, see section 19.3.

E442

[1SW]Prerequisites:4,3, Section  19.3.

Given \(A,C∈ ℂ^{n× n}\) and \(F:ℝ→ℂ^{n× n}\) continuous matrix valued functions, solve the ODE

\[ X'=AX+F~ ~ , X(0)=C~ ~ , \]

where \(X:ℝ→ℂ^{n× n}\).

( Hint: use the method of variation of constants: replace \(Y(t)=\exp (-tA)X(t)\) )

Hidden solution: [UNACCESSIBLE UUID ’1SX’]

E442

[1SY]Prerequisites:4,3, Sec. 19.3.Difficulty:*.

Given matrixes \(A,B,C∈ ℂ^{n× n}\), solve the ODE

\[ X'=AX+XB~ ~ , X(0)=C~ ~ , \]

where \(X:ℝ→ℂ^{n× n}\)

Hidden solution: [UNACCESSIBLE UUID ’1SZ’]

[UNACCESSIBLE UUID ’1T0’]

[1T1] Prerequisites:2,2.Difficulty:*.

Let \(V=ℂ^{n× n}\) a matrix space, we equip it with a submultiplicative norm \(\| C\| _ V\). Let \(C∈ V\) and let \(A,B:ℝ→ V\) be continuous curves in space of matrices.

  • We recursively define \(Q_ 0=C\), and

    \[ Q_{n+1}(s)=∫_ 0^ s A(𝜏) Q_ n(𝜏)B(𝜏)\, {\mathbb {d}}𝜏\quad ; \]

    show that the series

    \[ Y(t)=∑_{n=0}^∞ Q_ n(t) \]

    is well defined, showing that, for every \(T{\gt}0\), it converges totally in the space of continuous functions \(C^ 0=C^ 0([-T,T]→ V)\), endowed with the norm

    \[ \| Q\| _{C^ 0}{\stackrel{.}{=}}\max _{|t|≤ T} \| Q(t)\| _ V \quad . \]

  • Show that the function just defined is the solution of the differential equation

    \[ \frac{d\hskip5.5pt}{d{t}} Y(t) = A(t) Y(t) B(t)~ ~ ~ ,~ ~ ~ Y(0)=C~ ~ . \]

  • If \(A,B\) are constant, note that

    \[ Y(t)=∑_{n=0}^∞ t^ n \frac{A^ n C B^ n}{n!}\quad . \]

Hidden solution: [UNACCESSIBLE UUID ’1T2’] [1T3] Prerequisites:2,3.Note:Abel’s identity.

Let be given \(C∈ ℂ^{n× n}\), \(A:ℝ→ℂ^{n× n}\) continuous, and the solution \(Y(t)\) of the ODE

\[ \frac{d\hskip5.5pt}{d{t}} Y(t) = A(t) Y(t)~ ~ ~ ,~ ~ ~ Y(0)=C \]

(which has been studied in 2). Set \(a(t)={\operatorname {tr}}(A(t))\), show that

\[ \det (Y(t)) = \det (C) e^{∫_ 0^ t a(𝜏)\, {\mathbb {d}}𝜏 } \quad . \]

If \(C\) is invertible, it follows that \(Y(t)\) is always invertible.

Hidden solution: [UNACCESSIBLE UUID ’1T4’]

[UNACCESSIBLE UUID ’1T5’] [1T6]Prerequisites:4,3,2.

Let be given \(C∈ ℂ^{n× n}\), \(F,A:ℝ→ℂ^{n× n}\) continuous, and the solution \(Y(t)\) of the ODE

\[ \frac{d\hskip5.5pt}{d{t}} Y(t) = A(t) Y(t)~ ~ ~ ,~ ~ ~ Y(0)=\mathrm{Id}~ ~ . \]

Solve the equation

\[ X'=AX+F~ ~ , X(0)=C~ ~ , \]

where \(X:ℝ→ℂ^{n× n}\), using \(Y(t)\) as an auxiliary function.

Hidden solution: [UNACCESSIBLE UUID ’1T7’]