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THE VECPERMUTATION MATRIX THE VEC OPERATOR
JACOBIANS FOR TRANSFORMATIONS OF PATTERNED
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Aitken algorithm Anderson application Balestra Biometrics Unit Chapter commutation matrix commute covariance structural analysis crossed classifications defined definition denoted deriving Jacobians diagonal matrix dispersion matrices distinct eigenvalues eigenvalues equality relationships example expressions functionally independent elements functionally independent variables given gives hence Henderson and Searle identity matrix Jacobians for transformations JY-X Kronecker products Lancaster Lemma linear combinations linear model linear transformation MacRae Magnus and Neudecker Math matrix derivative matrix differentiation matrix equations matrix of order Moore-Penrose inverse multivariate analysis multivariate statistics Neudecker 1979 nonsingular notation order of Kronecker orthogonal partitioned matrix patterned matrices permutation matrix proof properties quadratic form quadratic subspace result Seely singular-value decomposition skew-symmetric spectral decomposition subscripts symmetric matrices Theorem Tracy and Dwyer Tracy and Singh values var(x'Ax var(x®x variance component models vec and vech vec operator vec-permutation matrix vecA vechX vecp vecpX vecX Wishart distribution x'Ax zero