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I've looked at this and it doesn't help because I don't know anything about SVD. Can someone dumb it down for me please?

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It is not exactly true that non-square matrices can have eigenvalues. Indeed, the definition of an eigenvalue is for square matrices. For non-square matrices, we can define singular values:

Definition: The singular values of a $m \times n$ matrix $A$ are the positive square roots of the nonzero eigenvalues of the corresponding matrix $A^{T}A$. The corresponding eigenvectors are called the singular vectors.

Of course, these have certain properties, that may or may not be useful for what you are trying to study.

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    $\begingroup$ Suggestion: change the first sentence for people who are looking for a quick answer and end up with inaccurate information. $\endgroup$ – Chris Anderson Mar 21 '17 at 21:16
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Eigenvalues and eigenvectors of a matrix, say $A$, help us find subspaces which are invariant under $A$ (when $A$ is seen as a linear transformation). If $A$ is non-square, then $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$, where $m\neq n$. Hence $Av=\lambda v$ makes no sense, since $Av\notin\mathbb{R}^m$.

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Non-square matrices do not have eigenvalues. If the matrix X is a real matrix, the eigenvalues will either be all real, or else there will be complex conjugate pairs.

Source: Wikipedia.

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