Question

how to find generalised eigen vector for matrix pair

Answer 1

In general, An eigenvector, v, of a matrix square A is a vector that satisfies the equation
\[Av = \lambda v\]
for some \lambda
That is
\[(A-\lambda I)v = 0\]
where I is the identity matrix

So the first step is the find the \lambda and you do that by finding a polynominal in \lambda from the determinant:

\[\det(A - \lambda I) = 0\]

Answer 2

It's kind of involved, especially for a site like this as it's hard to write out matrices efficiently. I am confident your text book will work a number of examples. Have a look at it.