eigenvalues and eigenvectors for rectangular matrices We can generalize matrix inverses from non-singular square matrices to rectangular matrices in general, for example, the well-known Moore–Penrose pseudoinverse. I am wondering how this can be done for eigenvalues and eigenvectors. 
Though $\det(|A-\lambda I|)=0$ cannot be used any more when $A$ is not square, there is nothing that prevents one to consider $Av=\lambda v$ for non-zero vector $v$ except the possibility of having an inconsistent linear system. 
Please give your comments and provide some references if there are some. 
Many thanks.
Edit
I know SVD. But it does not seem to be the one I wanted. For SVD of real matrix $A$, $A=UDV^T$ where $U, V$ are orthogonal matrices and $D$ is diagonal (with possibly zeros in the diagonal). We only have $AV_{*k}=\sigma_{k}U_{*k}$, $V_{*k}$ is the $k^\text{th}$ column of $V$. Since $V_{*k}$ and $U_{*k}$ are in general different, it does not resemble $Av=\lambda v$ for non-zero vector $v$ in the definition of eigenvectors. Also, even if we can have $A^TAV_{*k}=\lambda V_{*k}$, but this is for the (square) matrix $A^TA$, rather than $A$ itself.

Rectangular matrices do not have eigenvectors/eigvalues. Instead they have singular values & singular vectors. (Square matrices have both - eigenvalues and singular values). 
However, there is a simple relationship between eigenvalues and the singular values.
(Singular-values of matrix $A$) == sqroot (eigValues of $A^TA$). $A^TA$ has  have eigenvalues as it will be a square matrix irrespective of the dimensions of A.
This can be seen below: 
$ A = UDV^T ; A^T = VDU^T $
(Here, D is a diagonal matrix with singular values along its diagonal)
$ A^T*A = VDU^T * UDV^T $
As U and V are orthogonal matrices, the following relation holds:
$ U^TU = I and V^TV = I $
=> $ A^TA = VD^2V^T $
=> $ A^TA * (V^T)^{-1} = VD^2V^T * (V^T)^{-1} $
=> $ A^TA(V) = VD^2 $
Here, the diagonal values of $D^2$ are the eigenvalues of $A^TA$, which is the square of the singular values of matrix $A$. 
Additional Source: http://www-users.math.umn.edu/~olver/num_/lnv.pdf 
