By Peter J. Olver
Numerical computations in linear algebra (gaussian removing, eigenvalues, singular values, LU, SVD, etc.), differential equations, warmth and wave equations, approximation and interpolation, finite aspect technique. For the most recent model, see: http://www.math.umn.edu/~olver/num.html
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2. Since 1 2 −1 3 4 −3 1 2 1 1 −2 2 1 4 6 −5 1 0 0 3 4 −5 1 2 −1 −1 = 0 1 0 = 1 1 −1 −3 1 2, −7 0 0 1 4 6 −7 −2 2 1 1 2 −1 3 4 −5 we conclude that when A = −3 1 2 , then A−1 = 1 1 −1 . Observe that −2 2 1 4 6 −7 −1 there is no obvious way to anticipate the entries of A from the entries of A. 3. Let us compute the inverse X = 2 × 2 matrix A = a b . The right inverse condition c d AX = 5/18/08 y , when it exists, of a general w ax+ bz cx+ dz ay +bw cy + dw 40 = 1 0 0 1 = I c 2008 Peter J.
4), this is also the solution to the original system of linear equations, as you can check. 5). And that, barring a few minor complications that can crop up from time to time, is all that there is to the method of Gaussian Elimination! It is extraordinarily simple, but its importance cannot be overemphasized. Before exploring the relevant issues, it will help to reformulate our method in a more convenient matrix notation. 2. Gaussian Elimination — Regular Case. With the basic matrix arithmetic operations in hand, let us now return to our primary task.
The nonzero off-diagonal entries lij for i > j appearing in L prescribe the elementary row operations that bring A into upper triangular form; namely, one subtracts lij times row j from row i at the appropriate step of the Gaussian Elimination process. 5/18/08 51 c 2008 Peter J. Olver In practice, to find the L U factorization of a square matrix A, one applies the regular Gaussian Elimination algorithm to reduce A to its upper triangular form U . The entries of L can be filled in during the course of the calculation with the negatives of the multiples used in the elementary row operations.