By the fundamental theorem of invertible matrices, the null space of an invertible matrix has only the zero vector in it (it is trivial). Therefore there are no eigenvectors for the matrix. Is this correct? Are there any situations in which an invertible matrix has an eigenvalue?
Also, something I noticed is that even non-invertible matrices can not have any eigenvalues. So the invertibility of a matrix tells us it has no eigenvalues, but a lack of invertibility tells us nothing. Is this correct?