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  2. Gaussian elimination - Wikipedia

    en.wikipedia.org/wiki/Gaussian_elimination

    A variant of Gaussian elimination called Gauss–Jordan elimination can be used for finding the inverse of a matrix, if it exists. If A is an n × n square matrix, then one can use row reduction to compute its inverse matrix, if it exists. First, the n × n identity matrix is augmented to the right of A, forming an n × 2n block matrix [A | I].

  3. Tridiagonal matrix algorithm - Wikipedia

    en.wikipedia.org/wiki/Tridiagonal_matrix_algorithm

    In numerical linear algebra, the tridiagonal matrix algorithm, also known as the Thomas algorithm (named after Llewellyn Thomas), is a simplified form of Gaussian elimination that can be used to solve tridiagonal systems of equations. A tridiagonal system for n unknowns may be written as

  4. 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):

  5. Row echelon form - Wikipedia

    en.wikipedia.org/wiki/Row_echelon_form

    The reduced row echelon form of a matrix is unique and does not depend on the sequence of elementary row operations used to obtain it. The variant of Gaussian elimination that transforms a matrix to reduced row echelon form is sometimes called Gauss–Jordan elimination. A matrix is in column echelon form if its transpose is in row echelon form.

  6. Eigendecomposition of a matrix - Wikipedia

    en.wikipedia.org/wiki/Eigendecomposition_of_a_matrix

    Once the eigenvalues are computed, the eigenvectors could be calculated by solving the equation (), = using Gaussian elimination or any other method for solving matrix equations. However, in practical large-scale eigenvalue methods, the eigenvectors are usually computed in other ways, as a byproduct of the eigenvalue computation.

  7. Elementary matrix - Wikipedia

    en.wikipedia.org/wiki/Elementary_matrix

    Elementary row operations are used in Gaussian elimination to reduce a matrix to row echelon form. They are also used in Gauss–Jordan elimination to further reduce the matrix to reduced row echelon form .

  8. System of linear equations - Wikipedia

    en.wikipedia.org/wiki/System_of_linear_equations

    The standard algorithm for solving a system of linear equations is based on Gaussian elimination with some modifications. Firstly, it is essential to avoid division by small numbers, which may lead to inaccurate results. This can be done by reordering the equations if necessary, a process known as pivoting.

  9. Linear algebra - Wikipedia

    en.wikipedia.org/wiki/Linear_algebra

    Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy. [5] In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb.