Limit of a function with a matrix exponential

Question

I spent too many time trying to solve this problem...and finals are coming. Please help me!

I just can't see a method to do this demonstration:

"For an $A_{n \times n}$ matrix, demonstrate that a function f:$R^n$\{0} $\rightarrow$ $R$ defined by

\begin{eqnarray} f(x)=lim_{t\rightarrow \infty} \frac{1}{t}log\|e^{At}x\| \end{eqnarray}

can take at most, $n$ distinct values."

It seems to me that it's related to eigenvalues.

I tried to use $e^{At}=\sum_{k=0}^\infty \frac{t^k}{k!}A^k$, and $e^{At}=Se^JS^{-1}$, but I didn't get it out.

What would you do?

Answer 1

it is eigenvalue with the largest real part assuming that the matrix $A$ is diagonalizable. here is the reason. let $\{u_1, u_2, \cdots \}$ is a basis made up of eigenvectors of $A$ arranged such that $re(\lambda_1) > re(\lambda_2) \ge \cdots$.

suppose $x(0) = a_1u_1 + a_2u_2 + \cdots.$ then $x(t) = a_1e^{\lambda_1t} + a_2e^{\lambda_2 t} + \cdots.$

computing $|x|^2 = |a_1|^2 e^{2re(\lambda_1)} + \cdots$ and now taking the log gives you

$$\lim_{t \to \infty}\dfrac{|x(t)|}{t} = re(\lambda_1) $$

Answer 2

If the matrix is diagonalizable, then this reduces to a simple matter of going into the eigenspace and proving that in the limit, the following is true:

$$\lim_{t\to \infty}\frac{1}{t}\log (\sum_i a_i e^{\lambda_i t})=\max(\lambda_i)$$ where $\max$ selects from those $i$ that have $a_i> 0$.