The Space $$\mathcal{P}_{n}$$ of Positive n × n Matrices

Abstract

***our universe is a higher rank symmetric space P, the ***space of positiven x n real matrices:\r\r***The universe which We now enter is one of ***dimension at least 6 \rand this means that we will have trouble drawing meaningful pictures, \rnot to mention keeping our calculations on small pieces of paper. \r\rThe reader is advised to get some big sheets of paper to do some of the exercises.\r\r**Recall that we can identify SP2 with the Poincaré* upper half plane H (see Exercise 3.1.9 on p. 154 of Vol. I). \r\r**Our goal in this chapter is to extend to P, as many of the results of Chapter 3, Vol. I, as possible. \r\rFor example, we will study analogues of our favorite special functions-gamma, K-Bessel, and spherical. \r\r***The last two functions will be eigenfunctions of the Laplacian on P. They will display a bit more complicated structure than the functions we saw in Vol. I, but they and related functions have had many applications. \r\rFor example, James 328-330) and others have used ****Zonal polynomials and hypergeometric functions of matrix argument to good avail in multivariate statistics (see also Muirhead 468). \r\r\r***There are, in fact, many applications of analysis on P in multivariate statistics, which is concerned with data on several aspects of the same individual or entity; \re.g., reaction times of one subject to several stimuli. Multivariate statistics originated in the early part of the last century with Fisher and Pearson.\r\rIn Section 1.1 we will consider a very simple application of one of our coordinate systems for P, in the study of partial correlations, with an example from agriculture**.\r\r**Limit theorems for products of random matrices are also of interest; e.g., in demography (see Cohen 114). One can say something about this subject as well, by making use of harmonic analysis on P.***\r\r\r***We will see that P is a symmetric space;***\r i.e., a Riemannian manifold with a geodesic-reversing isometry at each point. \r\r\rWe will see that P is a symmetric space; i.e., a Riemannian manifold with a geodesic-reversing isometry at each point. \r\rMoreover it is a homogeneous space of the Lie group GL(n, IR)-the general linear group of all n x n non-singular real matrices. \r\rThe definition of Lie group is given around formula (2.1) of Chapter 2. \r\rBy homogeneous space, we essentially refer to the identification of P with the quotient KWG in Exercise 1.1.5 below. \r\rBecause P, is a symmetric space, harmonic analysis on P will be rather similar to that on the spaces considered in Volume I-R", S, and H. \r\rFor example, the ring of G-invariant differential operators on P is easily shown to be commutative from the existence of a geodesic-reversing isometry. \r\r\rThere are four main types of symmetric spaces: compact Lie groups, quotients of compact Lie groups, quotients of noncompact semisimple\r\r