By Olver P.J.

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It has the same number of rows as columns. A column vector is a m × 1 matrix, while a row vector is a 1 × n matrix. As we shall see, column vectors are by far the more important of the two, and the term “vector” without qualification will always mean “column vector”. A 1 × 1 matrix, which has but a single entry, is both a row and a column vector. The number that lies in the ith row and the j th column of A is called the (i, j) entry of A, and is denoted by aij . The row index always appears first and the column index second.

10) then multiplication by E1 will subtract twice the first row from the second row, multiplication by E2 will subtract the first row from the third row, and multiplication by E 3 will add 21 the second row to the third row — precisely the row operations used to place our original system in triangular form. Therefore, performing them in the correct order (and using the associativity of matrix multiplication), we conclude that when     1 2 1 1 2 1 then E3 E2 E1 A = U =  0 2 −1  . 11) A = 2 6 1, 5 0 0 1 1 4 2 The reader is urged to check this by directly multiplying the indicated matrices.

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.  0 0 −1  1 0 0 The corresponding lower triangular matrix is L =  2 1 0 ; its entries lying below 1 −1 1 the main diagonal are the negatives of the multiples we used during the elimination procedure.

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

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