Finding top eigenvector

How do you find the top eigenvector of a square matrix?

Solution

The most simple answer to this is the power method. It is a very simple algorithm that applies to any diagonalizable matrix. Diagonalizable matrices admit eigendecompositions, and thus powers $A^k$ will have eigenvalues $\lambda_i^k$. Hence the leading eigenvalue will tend to dominate as we increase $k$, and so as $k$ increases, $u_k$ will converge to the top eigenvector. Now that we have some intuition, here is the algorithm. For a matrix $A$, begin with a random unit vector $u_0$. The update rule is then: \(u_{k+1} = \frac{Au_k}{\|Au_k\|_2}\)

A more advanced method is Arnoldi iteration. Intuitively, it extends the idea of the power method, utilizing the information in $u_0,\dots,u_K$ to estimate the top $K$ eigenvalues of $A$. The full details are rather involved, and so I will simply advise interested readers to look into Arnoldi iteration themselves.