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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 ...
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. For such matrices, the half-vectorization is sometimes more useful than the ...
Input: initial guess x (0) to the solution, (diagonal dominant) matrix A, right-hand side vector b, convergence criterion Output: solution when convergence is reached Comments: pseudocode based on the element-based formula above k = 0 while convergence not reached do for i := 1 step until n do σ = 0 for j := 1 step until n do if j ≠ i then ...
In numerical linear algebra, the Gauss–Seidel method, also known as the Liebmann method or the method of successive displacement, is an iterative method used to solve a system of linear equations.
Starting with an initial guess x 0, this means we take p 0 = b − Ax 0. The other vectors in the basis will be conjugate to the gradient, hence the name conjugate gradient method. Note that p 0 is also the residual provided by this initial step of the algorithm. Let r k be the residual at the kth step:
where R 1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q 1 is m×n, Q 2 is m×(m − n), and Q 1 and Q 2 both have orthogonal columns. Golub & Van Loan (1996 , §5.2) call Q 1 R 1 the thin QR factorization of A ; Trefethen and Bau call this the reduced QR factorization . [ 1 ]
1 0 0 0 row k * * 1 0 * * * 1 col l^ Wqwt 18:48, 13 March 2018 (UTC) You are right. But it is not so easy, because, in the case singular matrices you must have two different variables, one for the pivot row and the other for the pivot column (here, both are called k). Moreover, the stopping test is when one of these two variables reaches its ...