# Exponential matrix problem and Putzer's formula

1. Jun 18, 2011

### kamenoss

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

$e^{ \begin{bmatrix} \lambda_{1} & 0 \\ 1& \lambda_{2} \end{bmatrix}t }$=\begin{bmatrix} e^{\lambda_{1}t} & 0 \\
\frac{e^{\lambda_{2}t} - e^{\lambda_{1}t}}{ \lambda_{2} - \lambda_{1}} & e^{\lambda_{2}t}
\end{bmatrix}

with $\lambda_{1}\neq \lambda_{2}$
2. Relevant equations

Could someone solve this for me?

3. The attempt at a solution

I am no good at maths... with basic knowledge of linear algebra.
It looks like $\lambda_{1} , \lambda_{2}$ are the eigenvalues of a matrix A that solves a differential system. All indications are pointing to Putzer's formula, but everything i have tried failed. Probably i am missing something....

p.s.Sorry for my terrible English

2. Jun 18, 2011

### hunt_mat

Basically you write out the definition of the exponential:
$$e^{x}=1+x+\frac{x^{2}}{2!}+\cdots +\frac{x^{n}}{n!}+\cdots$$
To compute the LHS of your equation, you will have you figure out an equation for:
$$\left( \begin{array}{cc} \lambda_{1} & 0 \\ 1 & \lambda_{2} \end{array}\right)^{n}$$
and this will give you what you require.