Zero as an element of an eigenvector

[ekkilop]

Quick question on eigenvectors;

Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector?

Thank you!

 

[AlephZero]

Suppose the zero is the last element of the eigenvector. Then you can partition the matrix and write
$\begin{bmatrix} A & u \\ v^T & s \end{bmatrix}\begin{bmatrix} x \\ 0 \end{bmatrix} = \lambda \begin{bmatrix} x \\ 0 \end{bmatrix}$ where $u$ and $v$ are vectors and $s$ is a single matrix element.

Multiplying out you get the two equations $Ax = \lambda x$ and $v^t x = 0$. Interestingly, the last column of the original matrix doesn't appear in those equations.

 

[ekkilop]

Thank you for your reply!

Is there a way to determine from the matrix whether a zero will appear without calculating the eigenvector explicitly?