dominant eigenvalue

If you take almost any starting vector and keep multiplying it by A over and over, the direction it settles into is the eigenvector of the largest-magnitude eigenvalue. That eigenvalue is the dominant eigenvalue: the one whose stretching outpaces all the others, so after enough multiplications it drowns them out and dictates the limiting direction.

Here is why. Expand the starting vector in an eigenbasis, x = c_1 v_1 + ... + c_n v_n. Then A^k x = c_1 lambda_1^k v_1 + ... + c_n lambda_n^k v_n. If |lambda_1| > |lambda_j| for all other j, the term lambda_1^k v_1 grows fastest, so dividing by its size leaves A^k x pointing essentially along v_1. The ratio of successive lengths converges to |lambda_1|. This is the power method, the simplest eigenvalue algorithm.

The dominant eigenvalue is the seed of huge applications. Google's original PageRank is the dominant eigenvector of the web's link matrix, computed by exactly this repeated-multiplication idea. The method needs a strict gap (|lambda_1| > |lambda_2|) and a starting vector with nonzero component along v_1; ties or complex dominant pairs need refinements, but the core intuition — biggest eigenvalue wins the long run — is exactly the spectral radius story made dynamic.