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A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions

Abstract

We provide a new, concise derivation of necessary and sufficient conditions for a real matrix-normally distributed matrix X and we characterize the general matrix-normal covariance structure such that, given the symmetric positive-semidefinite matrices Ai, i = 1,2,...,m, the matrix quadratic forms \({\textbf {X}}^{\prime }{\textbf {A}}_{i}{\textbf {X}}\) are distributed as independent noncentral Wishart matrices.

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Correspondence to Phil D. Young.

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Young, P.D., Patrick, J.D. & Young, D.M. A Brief Derivation of Necessary and Sufficient Conditions for a Family of Matrix Quadratic Forms to Have Mutually Independent Non-Central Wishart Distributions. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00260-5

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Keywords and phrases

  • Matrix trace
  • Kronecker product
  • chi-square random variable
  • matrix equations
  • generalized inverse matrix

AMS (2000) subject classification

  • Primary 15A09
  • Secondary 15A52