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How does Mathematica 9 assign eigenvectors to a singular matrix?

Posted 10 years ago
How does Mathematica assign the eigenvectors to a singular matrix? For example, the three-dimensional null matrix
mat = {{0, 0, 0}, {0, 0, 0}, {0, 0, 0}}

has the eigenvectors
In[4]:= Eigenvectors[mat]
Out[4]= {{0, 0, 1}, {0, 1, 0}, {1, 0, 0}}

What algorithm is used to order these degenerate eigenvectors?

I ask because in other situations it is not always anticipated that there will be degeneracies and hence singularities of the matrix.
POSTED BY: Bruce Shore
3 Replies
Posted 10 years ago
On the web page for Eigenvectors http://reference.wolfram.com/mathematica/ref/Eigenvectors.html click on "Details and Options" part way down the page and read all the information that then displays. I believe that might answer your questions.

I suspect that far too many users, particularly new users, do not realize they should almost automatically click on "Details and Options" on every help page they look at. Perhaps "Details and Options" should be blinking red with a little dancing finger beside this pointing at it with a little thought bubble saying "click me, click me."
POSTED BY: Bill Simpson
Mathematica will compute a null space corresponding to the (multiple) eigenvalue obtained from the characteristic polynomial. This is done via row reduction.
POSTED BY: Daniel Lichtblau
Maybe you should split "Details and Options" into 2 sections:  "Options" and "How It Works"
POSTED BY: Frank Kampas
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