math2J winter 2010 CLASS NOTES

math2J winter 2010 CLASS NOTES - A is A = LU where L is a...

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Class Notes from January 20, 2010 Today we covered Section 1.5. We did an example of inverting a matrix using elementary matrices and ended with the triangular decomposition. The first topic was the computation of A - 1 when A is a nonsingular matrix. The inverse can be expressed as the product of elementary matrices. Remember the following equivalent statements: (a) A is nonsingular (b) the equation Ax = 0 has only x = 0 as solution (c) A is row equivalent to the identity matrix Statement (c) means there are elementary matrices E 1 , E 2 , ...,E k such that E k E k - 1 ··· E 1 A = I. This means A - 1 = E k E k - 1 ··· E 1 . This observation leads to a way to compute A - 1 which we did in class: ( A | I ) ( E 1 A | E 1 I ) ( E 2 E 1 A | E 2 E 1 I ) · · ( E k E k - 1 ··· E 1 A | E k E k - 1 ··· E 1 I ) . At the last step, E k E k - 1 ··· E 1 A = I and E k E k - 1 ··· E 1 I = E k E k - 1 ··· E 1 gives A - 1 . The triangular decomposition of
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Unformatted text preview: A is A = LU where L is a lower triangular matrix and U is upper triangular. This was done by multiplying elementary matrices times A until the result is an upper triangular matrix U, E k E k-1 E 1 A = U. Then we multiply the inverses of the elementary matrices to solve for A, A = E-1 1 E-1 1 E-1 k U. It turns out that L = E-1 1 E-1 1 E-1 k is a lower triangular matrix. This is because the elementary row operations we used to nd a row equivalent upper triangular matrix to A are lower triangular and the product of lower triangular matrices is lower triangular and the inverse of a lower triangular matrix is lower triangular. 1...
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