# Exponential matrix proof

I have to prove the inverse of the matrix e^A. we haven't studied exponential matrices in uni but he gave us the definition of it with the series e^A= I+A+A^2 ... A^k where A^k=0, even for numbers greater than k.
I have tried to think of a way to prove it, but neither my classmates or I found something. I looked up google etc, but all the proofs were with things that we didn't learn. Any help is welcome.

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ShayanJ
Gold Member
Consider this:if $A={a_{ij}}$ then for $B=e^A$ we have $b_{ij}=e^{a_{ij}}$.

HallsofIvy
Homework Helper
Prove what? Not that e^A= I+ A+ A^2/2+ A^3/3!+ ... because you were given that as a definition. And not that $$A^n= 0$$ for n greater than some k because that is not generally true.

Perhaps your teacher was saying that if there exist some k, such that if A^n = 0 if n> k, then e^A= 1+ A+ A^2/2!+ A^3/3!+ ...+ A^k/k!

Either your teacher gave e^A= I+ A+ A^2/2+ A^3/3!+ ... (an infinite sum) from which it follows that if A^n= 0 for n> k that e^A= I+ A+ A^2/2+ A^3/3!+ ... + A^k/k! or your teacher, wishing to avoid the technical complications involved in defining infinite sums of matrices, just said that if A^n= 0 for n> k, then e^A= I+ A+ A^2/2+ A^3/3!+ ... + A^k/k!

In either case, it does not require "proof" because it is a definition.

Consider this:if $A={a_{ij}}$ then for $B=e^A$ we have $b_{ij}=e^{a_{ij}}$.
Yes but how does that come out from the definition which he gave? :\

@HallsofIvy :

http://en.wikipedia.org/wiki/Exponential_matrix#Nilpotent_case
He basically gave that case, and asked us to prove what the inverse will be.

Office_Shredder
Staff Emeritus
Consider this:if $A={a_{ij}}$ then for $B=e^A$ we have $b_{ij}=e^{a_{ij}}$.