In this paper, we consider the distribution of the maximum latent root of a certain positive definite symmetric random matrix. For this purpose, we give a useful transformation of a symmetric matrix and calculate its Jacobian. We also give some useful expansion formulas for zonal polynomials (A. T. James ).
Recently Sugiyama  and Sugiyama and Hukutomi  gave the density functions of maximum latent roots of a central Wishart matrix when the covariance matrix Σ=I, and of a multivariate Beta matrix and a multivariateF-matrix in the central case.
Here we derive the density functions of maximum latent roots of a multivariate non-central Beta matrix, a non-central Wishart matrix and a multivariate central quadratic form with the covariance matrix Σ, and we also derive the density function of maximum canonical correlation coefficient. The notations in this paper are due to A. T. James  and A. G. Constantine .
Density Function Covariance Matrix Latent Root Expansion Formula Joint Density Function
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