# Matrix exponential

BearY

## Homework Statement

Show that if ##λ##and ##V ## are a pair of eigenvalue and eigenvector for matrix A, $$e^Av=e^λv$$

## Homework Equations

##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n##

## The Attempt at a Solution

I don't know where to start.

Last edited:

Gold Member
I'd use a lower case ##v## or better ##\mathbf v## here. (I've never seen capitalized vectors?)

## Homework Equations

##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n##
The power series is quite instructive. Each side is some matrix that is a function of ##A##. What happens if you multliply each side by an eigenvector of A? The idea is there are a lot of moving parts here... if you can find a fixed point maybe it isn't so hard.
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btw, do you know why the series is convergent for matrices? This is a bit subtler of a question.

BearY
Homework Helper
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## Homework Statement

Show that if ##λ##and ##V ## are a pair of eigenvalue and eigenvector for matrix A, $$e^AV=e^λV$$

## Homework Equations

##e^A=\sum\limits_{n=0}^\infty\frac{1}{n!}A^n##

## The Attempt at a Solution

I don't know where to start.

Start with ##A V = \lambda V##. What are ##A^2 V##, ##A^3 V##, etc.?

BearY
Mentor
Thread locked as the OP has shown no effort.