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If $W$ is a continuously differentiable $n×n$ matrix valued function on the rectangle $J×J \subset \mathbf{R^2} $, how can I show that

$W(r,s)W(s,r) = I$

$W(r,s)W(s,t) = W(r,t) $

$ \forall$ $r,s,t \in \mathbf{R} $

I've got these identities to work with;

$ W'(s,t) = A(s)W(s,t) $

$\dot{W}(s,t) = - W(s,t)A(t)$

$W(s,s) = I $

$A$ is a continuous $n×n$ matrix valued function defined on $J \subset \mathbf{R} $

Seven Hells
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1 Answers1

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The above identities are generalized differential equations, look here. The solution for the $W(s,t)$ will be a path ordered exponential. There will be a function like $\Pi(r,s):=T(e^{\int_r^s A(t) dt})$ and it holds $\Pi(s,s) = I$.

kryomaxim
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  • Thanks for reply. I was looking at this question http://math.stackexchange.com/questions/1139287/showing-ax-yay-z-ax-y-for-matrix-valued-function-a-with-following-proper?rq=1 Could you explain the last line to me, the "always replace" part? Thanks – Seven Hells Feb 08 '15 at 20:55