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Matrix normal distribution and elliptic distribution

Haoming Wang

Abstract

In this paper, we introduce the matrix normal distribution according to the tensor decomposition of its covariance. Based on the canonical diagonal form, the moment generating function of sample covariance matrix and the distribution of latent roots are explicitly calculated. We also discuss the connections between matrix normal distributions, elliptic distributions, and their relevance to multivariate analysis and matrix variate distributions.

Matrix normal distribution and elliptic distribution

Abstract

In this paper, we introduce the matrix normal distribution according to the tensor decomposition of its covariance. Based on the canonical diagonal form, the moment generating function of sample covariance matrix and the distribution of latent roots are explicitly calculated. We also discuss the connections between matrix normal distributions, elliptic distributions, and their relevance to multivariate analysis and matrix variate distributions.

Paper Structure

This paper contains 8 sections, 11 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Zonal polynomial
  3. Hypergeometric function
  4. Canonical diagonal form
  5. Covariance matrix
  6. Moment generating function
  7. Distribution of latent roots
  8. Conclusion

Key Result

Theorem 2.1

Let $C_{\kappa}(X)$ be a homogeneous symmetric polynomial with leading term $y_{1}^{k_{1}}\dots y_{m}^{k_{m}}$, which satisfies (eq: orthogonal invariant). The following statements are all equivalent.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Lemma 3.1: Constantine, 1963constantine1963some
  • proof
  • Definition 4.1: Matrix normal distribution
  • Theorem 4.2
  • proof
  • Theorem 4.3
  • Theorem 5.1: Dykstra, 1970dykstra1970establishing
  • proof
  • ...and 12 more