Gaussian Elimination as a Matrix Factorization Each of the elementary row operations from Gaussian Elimination (GE) has associated with it a nonsingular matrix with the property that multiplying (on the left) by that associated matrix gives the same result as applying the row operation. 1. Row operation 1 (R1): Multiply row i by a scalar α 6 = 0. Associated matrix: Let D be the identity matrix except for the ( i,i ) element, which is α . Then the matrix DA is the result of R1 applied to A . 2. Row operation 2 (R2): Interchange row i and row j . Associated matrix: Let P be the identity matrix with rows i and j interchanged. Then the matrix PA is the result of R2 applied to A . 3. Row operation 3 (R3): Multiply row j by a scalar m and add it to row i . Associated matrix: Let M be the identity but with m replacing the 0 in the ( i,j ) position. Then M = I + me i e t j , and the matrix MA is the result of R3 applied to A . Taking m = m ij =-a ij /a jj puts a zero in the ( i,j ) position of
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