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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.
If Gaussian elimination applied to a square matrix A produces a row echelon matrix B, let d be the product of the scalars by which the determinant has been multiplied, using the above rules. Then the determinant of A is the quotient by d of the product of the elements of the diagonal of B : det ( A ) = ∏ diag ( B ) d . {\displaystyle \det ...
Gaussian elimination is the main algorithm for transforming every matrix into a matrix in row echelon form. A variant, sometimes called Gauss–Jordan elimination produces a reduced row echelon form. Both consist of a finite sequence of elementary row operations; the number of required elementary row operations is at most mn for an m-by-n ...
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} .
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.
(Reuters) -U.S. President-elect Donald Trump's transition team is exploring ways to significantly reduce, merge, or even eliminate the top bank regulators in Washington, the Wall Street Journal ...
Dolphins don't score on 3 tries from Packers 1-yard line. It was 27 degrees at kickoff, with flurries, and the Dolphins appeared to freeze up, particularly in the first half, as they fell behind 24-3.
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.