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In mathematics, especially in linear algebra and matrix theory, the duplication matrix and the elimination matrix are linear transformations used for transforming half-vectorizations of matrices into vectorizations or (respectively) vice versa.
Notes Commutation matrix: The matrix of the linear map that maps a matrix to its transpose: See Vectorization: Duplication matrix: The matrix of the linear map mapping the vector of the distinct entries of a symmetric matrix to the vector of all entries of the matrix: See Vectorization: Elimination matrix
For a symmetric matrix A, the vector vec(A) contains more information than is strictly necessary, since the matrix is completely determined by the symmetry together with the lower triangular portion, that is, the n(n + 1)/2 entries on and below the main diagonal.
These decompositions summarize the process of Gaussian elimination in matrix form. Matrix P represents any row interchanges carried out in the process of Gaussian elimination. If Gaussian elimination produces the row echelon form without requiring any row interchanges, then P = I, so an LU decomposition exists.
During this process, we gradually modify the matrix using row operations until it becomes the matrix in which all the elements below the main diagonal are equal to zero. During this, we will simultaneously create two separate matrices P {\displaystyle P} and L {\displaystyle L} , such that P A = L U {\displaystyle PA=LU} .
Elimination theory culminated with the work of Leopold Kronecker, and finally Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which are described in the chapter on Elimination theory in the first editions (1930) of van der Waerden's Moderne Algebra.
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In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) = of certain algebraic groups = into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases.