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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]
A matrix is in reduced row echelon form if it is in row echelon form, with the additional property that the first nonzero entry of each row is equal to and is the only nonzero entry of its column. 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.
In particular, any two row equivalent matrices have the same row space. Any matrix can be reduced by elementary row operations to a matrix in reduced row echelon form. Two matrices in reduced row echelon form have the same row space if and only if they are equal. This line of reasoning also proves that every matrix is row equivalent to a unique ...
The row space is defined similarly. The row space and the column space of a matrix A are sometimes denoted as C(A T) and C(A) respectively. [2] This article considers matrices of real numbers. The row and column spaces are subspaces of the real spaces and respectively. [3]
In the case of matrices, the process involves manipulating either the rows or the columns of the matrix and so is usually referred to as row-reduction or column-reduction, respectively. Often the aim of reduction is to transform a matrix into its "row-reduced echelon form" or "row-echelon form"; this is the goal of Gaussian elimination.
Select the Kth row/column used to model the undesired internal nodes to be eliminated. Apply the below formula to all other matrix entries that do not reside on the Kth row and column. Then simply remove the Kth row and column of the matrix, which reduces the size of the matrix by one. Kron Reduction for the Kth row/column of an NxN matrix:
In practice, we can construct one specific rank factorization as follows: we can compute , the reduced row echelon form of .Then is obtained by removing from all non-pivot columns (which can be determined by looking for columns in which do not contain a pivot), and is obtained by eliminating any all-zero rows of .
A pivot position in a matrix, A, is a position in the matrix that corresponds to a row–leading 1 in the reduced row echelon form of A. Since the reduced row echelon form of A is unique, the pivot positions are uniquely determined and do not depend on whether or not row interchanges are performed in the reduction process. Also, the pivot of a ...