TITLE: A quick way to estimate eigenvector/eigenvalue of a matrix
QUESTION [8 upvotes]: Is there a quick way to give a raw estimation of an eigenvector/eigenvalue of a matrix? By "quick" I mean some method which can be computed without a computer or paper and pencil...something you could do in your head

REPLY [4 votes]: For a positive matrix, there is indeed a way to estimate a positive eigenvector from the matrix, the eigenvector associated to the eigenvalue described in the answer of Vladhagen.
To estimate a positive column eigenvector of a positive matrix, use the fact that the true ray of positive column eigenvectors is contained in the convex hull of the rays determined by the columns. So, for instance, any single column gives an estimate. Summing the columns gives, perhaps, a better estimate. From Vladhagen's second example I get the column sum
$$\begin{pmatrix}15\\11\\11\end{pmatrix}
$$
Now, one trouble with insisting on no calculation at all is that eigenvectors are usually given in normalized fashion, e.g. the $L^2$ normalization where you divide out by the $L^2$ norm which is the length of the vector. So I'll go on to do that with this example, getting
$$\begin{pmatrix} .694 \\ .509 \\ .509
  \end{pmatrix}
$$
This compares with the eigenvector calculated to 3 decimal places on some random on-line calculator that I found:
$$\begin{pmatrix} .698 \\ .483 \\ .528 \end{pmatrix}
$$
If you want a better eigenvector estimate, square the matrix before doing the calculation. Or cube it … As you take higher powers going to infinity, this is guaranteed to converge, by the Perron Frobenius Theorem, which is an example of what earlier comments call "iterative methods".