The Elimination Matrix: Some Lemmas and Applications
Jan R. Magnus, Heinz Neudecker · SIAM Journal on Algebraic and Discrete Methods · 1980
Two transformation matrices are introduced, L and D, which contain zero and unit elements only. If A is an arbitrary $( n,n )$ matrix, L eliminates from vecA the supradiagonal elements of A, while D performs the inverse transformation for symmetricA. Many properties of L and D are derived, in particular in relation to Kronecker products. The usefulness of the two matrices is demonstrated in three areas of mathematical statistics and matrix algebra: maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.