Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 2.3

Existence Theory

Mathematics — Calculus · matrix-exponential-properties

Problem

Let AA be an n×nn \times n matrix over R\mathbb{R}. Prove the following statements: (a) If AA is symmetric, then exp⁡At\exp At is symmetric. (b) If AA is skew-symmetric, then exp⁡At\exp At is orthogonal. (c) If λ∈R\lambda \in \mathbb{R} is an eigenvalue of AA with corresponding eigenvector x0∈Rnx_0 \in \mathbb{R}^n, then u(t)=etλx0u(t) = e^{t\lambda}x_0 is the solution of the initial value problem u′=Au,u(0)=x0.u' = Au, \quad u(0) = x_0. (d) If AA is block diagonal: A=[B1B2⋱Bk],A = \begin{bmatrix} B_1 & & \\ & B_2 & \\ & & \ddots \\ & & & B_k \end{bmatrix}, then exp⁡At=[exp⁡B1texp⁡B2t⋱exp⁡Bkt].\exp At = \begin{bmatrix} \exp B_1 t & & \\ & \exp B_2 t & \\ & & \ddots \\ & & & \exp B_k t \end{bmatrix}.

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