Eigenvalues of a rectangular matrix

Question

I've read that the singular values of a matrix are equal to the $$\sigma=\sqrt{\lambda_{K}}$$ where $\lambda$ are the eigenvalue but I'm assuming this only applies to square matrices. How could I determine the eigenvalues of a non-square matrix. Pardon my ignorance.

Answer 1

Eigenvalues aren't defined for rectangular matrices, but the singular values are closely related: The right and left singular values for rectangular matrix M are the eigenvalues of M'M and MM'.

Answer 2

The standard definition of an eigenvalue of $A$ is a number $\lambda$ so that for some vector $v$, $Av=\lambda v$. If you've got an $m$ by $n$ matrix, $v$ must be a vector of length $m$, and then $Av$ must be a vector of length $n$. Thus if $m\not= n$, there is no way for $\lambda v$ to be equal to $Av$, since the two vectors are of different dimensions.

As leshik suggests in the comments, we need to use an alternative definition if we want to look at the "eigenvalues" of a rectangular matrix. An analog is to find a unit vector $\hat{u}$ so that $Mv=\sigma u$ and $M^*u = \sigma v$. Then $u$ and $v$ are the left- and right-singular vectors for the value $\sigma$ (the analogs to an eigenvector of a square matrix).