The Telegraph Equation

[durandal]

Homework Statement

Find the general solution of the system of telegraphers equations using the eigenvalues and -vectors.

Relevant Equations

See image in description

I have tried row reduction to solve for the first eigenvector but I don't feel like I get any closer to the solution:

 

[DaveE]

Looks great so far, you just need to keep going.

An important thing to remember about eigenvectors is that there isn't a single vector solution. Eigenvectors are the vectors that satisfy ##Av=\lambda v## so any vector in the same direction as ##v## is also an eigenvector. This means that there is an unspecified degree of freedom that corresponds to the length of the eigenvectors. So, you get to arbitrarily choose one value. You're not solving for a vector, you're solving for a set of vectors of arbitrary magnitude. In practice we only pick and show one, which can confuse people.

ex. if your system has ##V_1=[1, -1]^T## as an eigenvector, then ##V_1=[-13, 13]^T## is also an eigenvector. We would normally say it's the same eigenvector, but that is sloppy shorthand for it's a member of that set of vectors (the eigenspace).

https://www.khanacademy.org/math/li...-finding-eigenvectors-and-eigenspaces-example

PS: Thank you for clearly stating the problem and your attempt. It makes it easy for us to understand and comment. If it's not easy to understand, people might not respond. This is a great example of how to ask a technical question.