  Moran's index is a basic measure of spatial autocorrelation, which has been
applied to varied fields of both natural and social sciences. A good measure
should have clear boundary values or critical value. However, for Moran's
index, both boundary values and critical value are controversial. In this
paper, a novel method is proposed to derive the boundary values of Moran's
index. The key lies in finding conditional extremum based on quadratic form of
defining Moran's index. As a result, two sets of boundary values are derived
naturally for Moran's index. One is determined by the eigenvalues of spatial
weight matrix, and the other is determined by the quadratic form of spatial
autocorrelation coefficient (-1<Moran's I<1). The intersection of these two
sets of boundary values gives four possible numerical ranges of Moran's index.
A conclusion can be reached that the bounds of Moran's index is determined by
size vector and spatial weight matrix, and the basic boundary values are -1 and
1. The eigenvalues of spatial weight matrix represent the maximum extension
length of the eigenvector axes of n geographical elements at different
directions. This work solves one of the fundamental problems of spatial
autocorrelation analysis.
