Let $x(t)=\exp(tA)\exp(tB)$ and $y(t)=\exp(t(A+B))$.
Show that if $AB=BA$ then $x(t)$ and $y(t)$ satisfy the same initial value problem for ODEs and therefore must be equal.
$A, B$ square matrices.
2026-05-15 10:41:35.1778841695
Commuting Exponential Matrices
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Hint: (1) What ODEs do $x$ and $y$ fulfill? Compute $x'$ and $y'$, try to find the given terms for $x$ and $y$ in your computed expressions. Write down a differential equation for $x$ and $y$? Are they equal?
(2) Compute $x(0)$ and $y(0)$. Are these values equal?