y b where U. matrix inverse A 1 ( L. 1 U 1. L 1 U 13 U 23 U 33 U 13 2 U 12 1

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1 LU decomposition -- manual demonstration Instructor: Nam Sun Wang lu-manualmcd LU decomposition, where L is a lower-triangular matrix with as the diagonal elements and U is an upper-triangular matrix Just as there are many combinations of, there are infinite number of combinations of L U However, when the diagonal elements of L are fixed to be, the remaining elements are uniquely fixed L U linear algebraic equation x b L U x b L y b where U x y matrix inverse ( L U) U L fter LU decomposition, we obtain solution x in a two-step process Step L U Step Solve L yb yl - b Step Solve U xy xu - y Example L U L L L work on the st row of b U U U U U U U U U U U U L L L work on the nd row of U U U L U L U L U U U U U U divide by! We stop here! U L L U For each row, there is a step where we divide by the diagonal element of If any of the diagonal element of is, LU decomposition does not exist Since which equation comes first makes no difference in the solution of x, we swap equations, which is equivalent to swapping rows of both and b U

2 lu-manualmcd Pivot Examine column # of all the rows in, the row with the largest element in this st column (in the absolute value sense) becomes the st row of the permutated matrix Likewise swapping for b Examine column # of all the rows from row# to the last row in, the row with the largest element in this nd column (in the absolute value sense) becomes the nd row of the permutated matrix nd so on L U L swap rows U U U U U b swap rows L L U If we work systematically from the first row of, we can solve for unknown elements in L and U matrices sequentially, each time with only one unknown work on the st row of U U U U U U L L L work on the nd row of U U U U U U L U L U L U U U L U L U U U L U L L work on the rd row of L U U U U U U L U L U b'

3 lu-manualmcd L U L U L U L U L U L U U U L U L U U U U hus, L L L L U U U U check: Step Solve L yb' yl - b' y y y Step Solve U xy xu - y x x x x x x x x b' y check: y b' y b' L y y b' L y L y y b' U x y y b' L y y b' L y L y x y U y U x U x U x y x U U x U x y U U x U x x y x U x x compare b compare b' Swapping rows of does not affect the answer x, as long as rows of b are also similarly swapped ( )

4 lu-manualmcd Mathcad's lu function returns matrices: P, L, U such that P L U P is a permutation matrix that has "" occupying some elements P i,j that signifies the raw swapping operation from row j to row i PLU lu( ) PLU P submatrix( PLU,,,, ) L submatrix( PLU,,,, ) U submatrix( PLU,,,, 9) P L 5 U Pre-multiplication by a permutation matrix row swapping he st row of P has P nd row in goes into st row in he nd row of P has P rd row in goes into nd row in he rd row of P has P st row in goes into rd row in hus, the permutated matrix has: row row row of check P compare L U P is orthonormal P P P P I P P P pplying P to the permutated matrix reverses the orignal permutation and yields back the original matrix 5 5 L 5 U P 5 5 compare U L Post-multiplication by a permutation matrix column swapping In the equation below, P - P is also a permutation matrix Post-multiplying of - by P - P has the following effect: he st column of P - has (P - ) nd column in - goes into st column in - he nd column of P - has (P - ) rd column in - goes into nd column in - he rd column of P - has (P - ) st column in - goes into rd column in - hus, the permutated matrix - has: column column column of - Swapping rows of results in swapping columns of - in the same order

5 Effect of swapping rows on matrix inverse compare P 5 I ( P )( P ) ( P ) P ( P ) P P P P 5 lu-manualmcd Post-multiplication by a permutation matrix column swapping In the equation above, post-multiplying of - by P has the following effect: he st column of P has (P) rd column in - goes into st column in - he nd column of P has (P) st column in - goes into nd column in - he rd column of P has (P) nd column in - goes into rd column in - hus, the permutated matrix - has: column column column of - From - to -, swap columns of - in a reverse order Post-multiplication by a permutation matrix column swapping In the equation below, post-multiplying of by P has the following effect: he st column of P has P rd column in goes into st column in ' he nd column of P has P st column in goes into nd column in ' he rd column of P has P nd column in goes into rd column in ' hus, the permutated matrix ' has: column column column of ' P '

6 lu-manualmcd Gaussian Elimination & LU Decomposition Let us illustrate with the same matrix and vector b as before Step ugment matrix and vector b b augment(, b ) b b We represent the steps Gaussian elimination takes in manipulating the elements in the augmented matrix b by pre-multiplying with a square matrix, which acts as an operator that operates on the second matrix Pivoting: swap st & nd eqn, because eqn () has the largest leading coefficient: P b' P b b' * () by / & subtract it from () () * () by / & subtract it from () () G Pivoting: swap nd & rd eqn: P () () () "" in the diagonal position for the st row of G means just transcribe the st row of b' and do nothing "-/" means subtract / of st row of b', and "" means add x of nd row of b' "-/" means subtract / of st row of b', and "" means add x of rd row of b' b' G b' b' * () by /(/) & subtract it from () () G 5 b' P b' b' b' G b' b' () () () () () () () () 5 () () () ()

7 lu-manualmcd Below is a minor variation of the above steps where we perform all the pivoting first, rather than pivoting as we go in each step combination of two sequential swapping steps is equivalent to pre-multiplying the augmented matrix b by P, which does multiple swappings in one sweep P P P P * () by / & subtract it from () () * () by / & subtract it from () () G b' G b' b' * () by /(/) & subtract it from () () G 5 b' G b' b' b' P b b' () () () () () () () () 5 () () () () We combine the two sequential Gaussian elimination steps G & G into an equivalent one single operation G: G G G G 5 b' G P b b' 5 8 he following play on math shows that since the ""matrix in b is upper triangular, the inverse of G is lower triangular and this is the L matrix hus, the lower triangular matrix L summarizes all the individual forward elimination steps taken during Gaussian elimination leading up to an upper triangular form, and Gaussian elimination is directly related to LU decomposition b' G P b G b' P b submatrix( b',,,, ) L b' P b L G L Check: L b' L 5 5 and U compare P b compare P