Question 122231
Although it's difficult to give a general answer, here goes. 
You do it the same way you solve an eigenvalue/eigenvector problem for a regular (non-parametric) matrix. 
Let's use an example.
Let A be your 2x2 parametric matrix,
{{{A=(matrix(2,2,1,s,3,0))}}}
where s is your parameter.
First, set up your eigenvalue problem,
{{{A-(sigma)I=(matrix(2,2,1-sigma,s,3,-sigma))}}}
I'm using {{{sigma}}} instead of lambda, the traditional eigenvalue Greek symbol, here.
Find your characteristic polynomial,
{{{(1-sigma)(-sigma)-3s=0}}}
{{{sigma^2-sigma-3s=0}}}
{{{sigma = (-(-1) +- sqrt( (-1)^2-4*1*(-3s)))/(2) }}} 
{{{sigma = (1 +- sqrt( 1+12s))/(2) }}} 
Now it gets tricky. 
As you can see from the solution for {{{sigma}}}, the parameter(s) position(s) in the parametric matrix determines what effect it has on the characteristic polynomial. 
In this example, depending on whether s is positive or negative, could lead to real or complex roots. 
Once you have the eigenvalues, you would go back to your matrix,
{{{(A-(sigma)I)(x)=(matrix(2,2,1-sigma,s,3,-sigma))*(matrix(2,1,x[1],x[2]))=(matrix(2,1,0,0))}}}
The solution vectors ({{{x[1]}}},{{{x[2]}}}) are the eigenvectors tied to each of the specific eigenvalues {{{sigma}}}. 
The parameter information is included through the characteristic polynomial for the eigenvalue and the matrix for the eigenvector.
Hope it helps!