The matrix definition of an eigen value is very useful since it allows us to find eigen values for a given matrix using the following theorem:
is an eigen value of iff
but since is non-zero we know that is singular, ie it's determinant is zero so an eigen value of A will satisfy the equation
which is known as the characteristic equation. (haven't proved the converse, but this is not required when calculating eigenvalues).
In the case is a matrix, this equation leads to the characteristic polynomial :
This is simply a quadratic equation and the roots of this are the eigen values of
In order to find the corresponding eigen vectors, we simply solve the equation
which will be two simultaneous equations. There will in fact be infinitely many solutions to this equation since any scalar multiple of an eigen vector is also an eigen vector.