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matrix - What does it mean when Mathematica returns a zero "eigenvector"?



For example, I ask Mathematica to compute the eigenvectors of $\left(\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & -4 \\ 0 & 0 & 4 & 5 \end{array} \right) $ and it returns {{1, 0, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 1}, {0, 0, 1, 0}}. What does this mean?



Answer



Notice the $2\times2$ block in the upper left portion of your matrix. This is what is known as a Jordan block. Jordan blocks are well-known to be defective; that is, they do not have a complete set of eigenvectors.


Eigenvectors[{{1, 1}, {0, 1}}]
   {{1, 0}, {0, 0}}


Since Mathematica is unable to yield a complete eigenvector set, it pads the list of eigenvectors with an appropriate number of $\mathbf 0$ vectors so that $n$ vectors are still returned.


In your case, you have the direct sum of a Jordan block and an ordinary matrix, and the defectiveness of this matrix is inherited from the Jordan block. That's why only three linearly independent vectors are returned, in addition to the padding.


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