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  2. Duplication and elimination matrices - Wikipedia

    en.wikipedia.org/wiki/Duplication_and...

    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.

  3. Vectorization (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Vectorization_(mathematics)

    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.

  4. List of named matrices - Wikipedia

    en.wikipedia.org/wiki/List_of_named_matrices

    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.

  5. Numerical linear algebra - Wikipedia

    en.wikipedia.org/wiki/Numerical_linear_algebra

    For many problems in applied linear algebra, it is useful to adopt the perspective of a matrix as being a concatenation of column vectors. For example, when solving the linear system =, rather than understanding x as the product of with b, it is helpful to think of x as the vector of coefficients in the linear expansion of b in the basis formed by the columns of A.

  6. Elimination theory - Wikipedia

    en.wikipedia.org/wiki/Elimination_theory

    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.

  7. LU decomposition - Wikipedia

    en.wikipedia.org/wiki/LU_decomposition

    The matrix () is the matrix in which the elements below the main diagonal have already been eliminated to 0 through Gaussian elimination for the first columns. Below is a matrix to observe to help us remember the notation (where each ∗ {\displaystyle *} represents any real number in the matrix):

  8. Matrix decomposition - Wikipedia

    en.wikipedia.org/wiki/Matrix_decomposition

    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.

  9. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    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 ...