# Finding the fundamental matrix where psi(0) = the identity matrix

1. Dec 6, 2011

### capertiller

1. The problem statement, all variables and given/known data

If I have a solution to a system of first order linear equations: $<x,y> = c_1 e^{-3t} <1,-1> + c_2 e^{-t} <1,1>$ , how do I find the fundamental matrix psi(t) so that psi(0) = I ?

2. Relevant equations

3. The attempt at a solution

$psi(t) = <<e^{3t}, e^{-t}>, <-e^{-3t}, e^{-t}>>$
$psi(0) = <<1, 1>, <-1, 1>>$

This is clearly not the identity matrix.
Now what?

2. Dec 6, 2011

### lanedance

just an idea but you have two linearly independent basis vectors call them u1, u2

could you take a linear comibination of your vectors such that
v1=au1+bu2 gives v1(0) = <1,0>
and
v2=cu1+du2 gives v2(0) = <0,1>?

then could you use those basis vectors to write the reuired fundamental matrix?