Identifying the largest component of the dominant eigenvector of a matrix

The largest eigenvalue in magnitude of an n x n matrix is called the dominant eigenvalue. Whenever this eigenvalue is simple it will have only one linearly independent eigenvector, called the dominant eigenvector. In many applications of linear algebra, the components of the dominant eigenvector are important, particularly the largest. First row dominance conditions which guarantee that a given component of the dominant eigenvector will have the largest magnitude are explored. Next two algorithms to compute the dominant eigenvector of non-negative matrices which obtain the dominant eigenvalue as well are developed. Convergence of one of these allgorithrms is proved. An example is provided which illustrates many of the theorems, and the algorithms are worked through on this example. A simple small-scale BASIC program for the second algorithm is provided, along with some conjectures found to be false and suggested topics for further investigatio