38
$\begingroup$

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?

$\endgroup$
36
$\begingroup$

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.

$\endgroup$
2
  • 8
    $\begingroup$ Suggestion: change the first sentence for people who are looking for a quick answer and end up with inaccurate information. $\endgroup$
    – chris
    Mar 21 '17 at 21:16
  • $\begingroup$ The co-kurtosis matrix is non-square. does it have eigenvalues and eigenvectors, or singular values and singular vectors? $\endgroup$
    – develarist
    Sep 9 '20 at 1:53
27
$\begingroup$

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 $Av\notin\mathbb{R}^m$.

$\endgroup$
3
  • 1
    $\begingroup$ Can someone interpret the notation A: R^m => R^n. Does it mean all the elements in A are real numbers ? $\endgroup$
    – Amey Yadav
    Jul 3 '20 at 3:12
  • $\begingroup$ Yes, it does. Of course we can consider matrices over $\mathbb{C}$ instead of the reals, etc $\endgroup$
    – AnyAD
    Jul 6 '20 at 12:02
  • $\begingroup$ @AmeyYadav If I'm not wrong, $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$ represents $A$ as a transformation from an $m$-dimensional space to an $n$-dimensional space. I believe answerer here is assuming that $A$ contains only real numbers. $\endgroup$
    – koyae
    Jan 14 at 3:39
8
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.