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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].
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.
Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix is first created with the left side being the matrix to invert and the right side being the identity matrix. Then, Gaussian elimination is used to convert the left side into the identity matrix ...
The Schur complement arises when performing a block Gaussian elimination on the matrix M.In order to eliminate the elements below the block diagonal, one multiplies the matrix M by a block lower triangular matrix on the right as follows: = [] [] [] = [], where I p denotes a p×p identity matrix.
Otherwise, the Bareiss algorithm may be viewed as a variant of Gaussian elimination and needs roughly the same number of arithmetic operations. It follows that, for an n × n matrix of maximum (absolute) value 2 L for each entry, the Bareiss algorithm runs in O( n 3 ) elementary operations with an O( n n /2 2 nL ) bound on the absolute value of ...
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.
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
Download as PDF; Printable version ... This is compared to the nine operations for the full 3x3 matrix. Another notable example of a separable filter is the Gaussian ...