AIMS lecture notes on numerical analysis by Olver P.J.

By Olver P.J.

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If the algorithm fails to be completed, which happens whenever zero appears in any diagonal pivot position, then the original matrix is not regular, and does not have an L U factorization.   2 1 1 5 2 . 4. Let us compute the L U factorization of the matrix A =  4 2 −2 0 Applying the Gaussian Elimination algorithm, we begin by adding −2 times the first row to the second  row, and then  adding −1 times the first row to the third. The result is 2 1 1 the matrix  0 3 0 . The next step adds the second row to the third row, leading   0 −3 −1 2 1 1 to the upper triangular matrix U =  0 3 0 , whose diagonal entries are the pivots.

4, we can readily solve any linear system with the given coefficient matrix by Forward and Back Substitution. For instance, to find the solution to      x 2 1 1 1    4  y = 2, 5 2 z 2 −2 0 2 we first solve the  1 0 2 1 1 −1 lower triangular system     1 0 a     0 b = 2, 2 1 c or, explicitly, a 2a + b = 1, = 2, a − b + c = 2. The first equation says a = 1; substituting into the second, we find b = 0; the final equation yields c = 1. We then use Back Substitution to solve the upper triangular system        2 x + y + z = 1, 1 a 2 1 1 x 0 3 which is 0y =  b = 0, 3y = 0, 1 c 0 0 −1 z − z = 1.

For example, BA may not be defined even when A B is. Even if both are defined, they may be different sized matrices. For example the product s = r c of a row vector r, a 1 × n matrix, and a column vector c, an n × 1 matrix with the same number of entries, is a 1 × 1 matrix or scalar, whereas the reversed product C = c r is an n × n matrix. For instance, (1 2) 3 0 = 3, 3 (1 2) = 0 whereas 3 0 6 0 . In computing the latter product, don’t forget that we multiply the rows of the first matrix by the columns of the second.

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