Matrix Exponential Homework: Eigenvalues & Eigenvectors

[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.

 

[StoneTemplePython]

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.

 

[Ray Vickson]

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.?

 

[Mark44]

Thread locked as the OP has shown no effort.