I don't need the template because this isn't a homework problem, per se; it's just information about how to get started on my homework.(adsbygoogle = window.adsbygoogle || []).push({});

So I have an equationx'=Ax,x(0)=x^{0}, where A is a constant matrix.

I can write it asx=ø(t)x^{0}, whereø(t) is a fundamental matrix such thatø(0)=I

We know that Taylor expanding e^{at}gives us ∑ a^{n}t^{n}/n!

(n starts from 1 and goes to infinity)

I+ ∑A^{n}t^{n}/ n! =I+At +A^{2}t^{2}/2! + ..... +A^{n}t^{n}/n! + ........ = e(At)

So, I don't even understand how you can raise a scalar to a power of a matrix. This is a messed-up world we live in.

d/dt e(At) = ∑A^{n}t^{n-1}/(n-1)! =A[I+ ∑A^{n}t^{n}/n!].

I don't understand that last step. Must be some property of summations?

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# Homework Help: Trying to follow my textbook's explanation

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