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Using row operations to convert a matrix into reduced row echelon form is sometimes called Gauss–Jordan elimination. In this case, the term Gaussian elimination refers to the process until it has reached its upper triangular, or (unreduced) row echelon form. For computational reasons, when solving systems of linear equations, it is sometimes ...
Row echelon form. In linear algebra, a matrix is in row echelon form if it can be obtained as the result of Gaussian elimination. Every matrix can be put in row echelon form by applying a sequence of elementary row operations. The term echelon comes from the French échelon ("level" or step of a ladder), and refers to the fact that the nonzero ...
LU decomposition can be viewed as the matrix form of Gaussian elimination. Computers usually solve square systems of linear equations using LU decomposition, and it is also a key step when inverting a matrix or computing the determinant of a matrix. The LU decomposition was introduced by the Polish astronomer Tadeusz Banachiewicz in 1938. [1]
These decompositions summarize the process of Gaussian elimination in matrix form. 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.
The use of Gaussian elimination for putting the augmented matrix in reduced row echelon form does not change the set of solutions and the ranks of the involved matrices. The theorem can be read almost directly on the reduced row echelon form as follows. The rank of a matrice is number of nonzero rows in its reduced row echelon form.
Bareiss algorithm. In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute the ...
Gram–Schmidt process. In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. By technical definition, it is a method of constructing an orthonormal basis from a set of vectors in an inner ...
Elementary matrix. In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation. The elementary matrices generate the general linear group GLn(F) when F is a field. Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right ...
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