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The Elimination Matrix: Some Lemmas and Applications

1980; Society for Industrial and Applied Mathematics; Volume: 1; Issue: 4 Linguagem: Inglês

10.1137/0601049

2168-345X

Jan R. Magnus, Heinz Neudecker,

Scientific Research and Discoveries

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.