  The leading eigenpair (the couple of eigenvalue and its eigenvector) or the
first nontrivial one has different names in different contexts. It is the
maximal one in the matrix theory. The talk starts from our new results on
computing the maximal eigenpair of matrices. For the unexpected results, our
contribution is the efficient initial value for a known algorithm. The initial
value comes from our recent theoretic study on the estimation of the leading
eigenvalues. To which we have luckily obtained unified estimates which consist
of the second part of the talk. In the third part of the talk, the original
motivation of the study along this direction is explained in terms of a
specific model. The paper is concluded by a brief overview of our study on the
leading eigenvalue, or more generally on the speed of various stabilities.
