The context for this is solving the gambler's ruin problem using linear algebra.
I haven't found a good explanation for why the linear combination of the eigenvalues for the matrix representing a system of difference equations yields solutions to the difference equation.
Reading this: Understanding why the roots of homogeneous difference equation must be eigenvalues
Confirmed some of what I thought. The matrix you are iterating does share the eigenvalues with whatever diagonal matrix it is similar to (somewhat tautological), but it still doesn't explain why the eigenvalues gives all (that it is exhaustive is especially important to me) solutions.
I have searched around pretty hard and have found the method but not the justification. Thanks.
