- #1
wuhtzu
- 9
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Homework Statement
[itex]\vec{N} = \frac{1}{\tau}(\int_0^\infty exp(t' \textbf{T}) dt') \vec{n}_0[/itex]
Show that [itex]\vec{N} = -\frac{\textbf{T}^{-1}}{\tau}\vec{n}_0[/itex] where [itex]\textbf{T}^{-1}[/itex] is the inverse matrix of [itex]\textbf{T}[/itex].
It is part of a small project called Gambler's Ruin in which we investigate the random walk of a gambler on a "money axis" where he has a probability for moving right and left. So the matrix T is the Laplacian matrix for a 1d graph (1)--(2)--(3)--(4)--(5)--(6)--(M).
Homework Equations
I have two main questions:
1. How to integrate the matrix
2. That [itex]\int_0^\infty exp(t) dt = \infty[/itex] as I expect this will be a part of the solution.
The Attempt at a Solution
First of all the matrix exponential is just the Taylor series of the exponential with the matrix as exponent:
[itex]exp(t\textbf{T}) = \sum_{k=0}^\infty \frac{1}{k!} (t\textbf{T})^k[/itex]
If the matrix T is diagonal then the exp(tT) will just be an matrix with exponential exp(t) along it's diagonal. If it is not diagonal all elementes will be proportinal to exp(xt).
So if this matrix T, the laplacian for a 1d graph, is diagonal (which I hope it is or can be made to be) I am stuck when I have to do the integral of a matrix with diagonal entries exp(t).
[itex]\frac{1}{\tau}(\int_0^\infty
\left[\begin{array}{ccc}
exp(t) & 0 & 0 \\
0 & exp(t) & 0 \\
0 & 0 & exp(t) \end{array}\right] dt') \vec{n}_0[/itex]
I'd be glad for any advice on how to proceed or hints to where I can find out about this.
Best regards
Wuhtzu