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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].
Simplified forms of Gaussian elimination have been developed for these situations. [ 6 ] The textbook Numerical Mathematics by Alfio Quarteroni , Sacco and Saleri, lists a modified version of the algorithm which avoids some of the divisions (using instead multiplications), which is beneficial on some computer architectures.
No (partial) pivoting is necessary for a strictly column diagonally dominant matrix when performing Gaussian elimination (LU factorization). The Jacobi and Gauss–Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant. Many matrices that arise in finite element methods are diagonally ...
This system has the exact solution of x 1 = 10.00 and x 2 = 1.000, but when the elimination algorithm and backwards substitution are performed using four-digit arithmetic, the small value of a 11 causes small round-off errors to be propagated. The algorithm without pivoting yields the approximation of x 1 ≈ 9873.3 and x 2 ≈ 4.
Before Gauss many mathematicians in Eurasia were performing and perfecting it yet as the method became relegated to school grade, few of them left any detailed descriptions. Thus the name Gaussian elimination is only a convenient abbreviation of a complex history. The Polish astronomer Tadeusz Banachiewicz introduced the LU decomposition in ...
The probability density function for the random matrix X (n × p) that follows the matrix normal distribution , (,,) has the form: (,,) = ([() ()]) / | | / | | /where denotes trace and M is n × p, U is n × n and V is p × p, and the density is understood as the probability density function with respect to the standard Lebesgue measure in , i.e.: the measure corresponding to integration ...
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) = of certain algebraic groups = into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases.
Gaussian algorithm may refer to: Gaussian elimination for solving systems of linear equations; Gauss's algorithm for Determination of the day of the week; Gauss's method for preliminary orbit determination; Gauss's Easter algorithm; Gauss separation algorithm
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