Notes on the Second Eigenvalue of the Google Matrix

If A is an n × n matrix whose n eigenvalues are ordered in terms of decreasing modules, |λ1| ≥ |λ2| ≥ · · · |λn|, it is often of interest to estimate |λ2| |λ1| . If A is a row stochastic matrix (so λ1 = 1), one can use an old formula of R. L. Dobrushin to give a useful, explicit formula for |λ2|. The purpose of this note is to disseminate these known results more widely and to show how they imply, as a very special case, some recent theorems of Haveliwala and Kamvar about the second eigenvalue of the Google matrix. If A = (aij) is an n × n real matrix, A has n (counting algebraic multiplicity) complex eigenvalues which can be listed in order of decreasing modules: |λ1| ≥ |λ2| ≥ · · · ≥ |λn|. We have |λ1| = sup{|λ| ∣ λ is an eigenvalue of A} and |λ1| is called the spectral radius of A, r(A) := |λ1|. In many problems it is of interest to estimate |λ2| |λ1| = |λ2| r(A) . Indeed, an analogous problem is of great interest for bounded linear maps on Banach spaces: see [2], [3] and the references there. Slightly more generally, suppose that V is an m-dimensional real vector space and L : V → V is a linear map. Again L has m possibly complex eigenvalues which can be written in order of decreasing modules: |λ1| ≥ |λ2| ≥ · · · ≥ |λm|. If ‖ · ‖ denotes any norm on V (recall that all norms on a finite dimensional real vector space give the same topology), we can define ‖L‖ = sup{‖Ly‖ : y ∈ V, ‖y‖ ≤ 1}. (1) It is known that r(L) = |λ1| = sup{|λ| : λ is an eigenvalue of L} (2) = lim k→∞ ‖L‖ 1 k = inf k≥1 ‖L‖ 1 k , where L denotes the composition of L with itself k-times. ∗Partially supported by NSF DMS-00-70829 1991 AMS Mathematics Subject Classification: Primary 15A18, 15A42, 15A48