Trying to follow my textbook's explanation

[Jamin2112]

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.

So I have an equation x'=Ax, x(0)=x⁰, where A is a constant matrix.

I can write it as x=ø(t)x⁰, where ø(t) is a fundamental matrix such that ø(0)=I

We know that Taylor expanding e^at gives us ∑ aⁿtⁿ/n!

(n starts from 1 and goes to infinity)

I + ∑Aⁿtⁿ / n! = I + At + A²t²/2! + ... + Aⁿtⁿ/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) = ∑Aⁿt^n-1/(n-1)! = A[ I + ∑Aⁿtⁿ/n!].

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

 

[boboYO]

write out some terms (say the first 3) instead of just ∑...

it should become clear.

 

[HallsofIvy]

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.

So I have an equation x'=Ax, x(0)=x⁰, where A is a constant matrix.

I can write it as x=ø(t)x⁰, where ø(t) is a fundamental matrix such that ø(0)=I

We know that Taylor expanding e^at gives us ∑ aⁿtⁿ/n!

(n starts from 1 and goes to infinity)

I + ∑Aⁿtⁿ / n! = I + At + A²t²/2! + ... + Aⁿtⁿ/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.

There is no "scalar to a power of a matrix" except on the far right. And that equation defines what is meant by e^A

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

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

\sum_{n=0}^\infty A^n t^{n-1}=A+ A^2t+ A^3t^2+ \cdot\cdot\cdot=A(I+ At+ A^2 t^2+ \cdot\cdot\cdot)= A(I+ \sum_{n=1}^\infty A^nt^n).