Linear Systems and Matrix Exponentials — Standard Exercises

date2026-07-16document_iddoc_2cdc4ad4fe7dad21bb2857a4065d4038description行列指数の時間発展、非対角行列の対角化、回転、一般高階方程式の一階化、非同次系の定数変化公式を確認する演習。prerequisites一階連立系と行列指数関数type問題演習content_typeexercisestatusactive

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1Problem 1: Composition of Time Evolution

Use uniqueness for initial-value problems to prove that $E (t) = e^{A t}$ satisfies $E (t + s) = E (t) E (s)$ . Also prove that $E (t)$ is invertible.

1.1Sample Answer

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As functions of $t$ , both $E (t + s)$ and $E (t) E (s)$ satisfy $X^{'} = A X$ and have the same value $E (s)$ at $t = 0$ . Uniqueness for the initial-value problem therefore gives

E (t + s) = E (t) E (s) .

Setting $s = - t$ yields

E (t) E (- t) = E (0) = I .

Consequently, $E (t)$ is invertible and

E (t)^{- 1} = E (- t) = e^{- A t} .