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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.
During execution of the Bareiss algorithm, every integer that is computed is the determinant of a submatrix of the input matrix. This allows, using the Hadamard inequality, to bound the size of these integers. Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic ...
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.
Pseudoinverse — a generalization of the inverse matrix. Row echelon form — a matrix in this form is the result of applying the forward elimination procedure to a matrix (as used in Gaussian elimination). Wronskian — the determinant of a matrix of functions and their derivatives such that row n is the (n−1) th derivative of row one.
Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes ...
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This lemma generalizes one of the Finsler's lemma variants by including an extra matrix C and an extra constraint involving this extra matrix. It is interesting to note that if the strict inequalities are changed to non-strict inequalities, the equivalence does not hold anymore: only the second statement imply the first statement.
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