Embarassing question about eigenvectors

[Ed Quanta]

Ok, so let us suppose we have a spinor which is a spin 1/2 state vector (a)
b

Now spinors exist in 2 dimensional complex space. How do I find the eigenvalues which correspond to the eigenvector
(a)
b


I am confused because we are dealing with eigenvalues for a matrix which is not a square matrix. I know for a square matrix we just find the eigenvalues such that the determinant of the matrix becomes zero. I am not sure how to deal with determinants of non-square matrices however. Helo anybody?

[Dr Transport]

in spin 1/2 space, the eigen vectors are usually found using the z component of the angular momentum. Use the Pauli matricies for S_{z} to find the eigenvalues and vectors.