Transcription of 11. LU Decomposition
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11. LU Decomposition
11. LU DecompositionCertain matrices are easier to work with than others. In this section, wewill see how to write any square matrixMas the product of two matricesthat are easier to work with. We ll writeM=LU, where: Lislower triangular. This means that all entries above the maindiagonal are zero. In notation,L= (lij) withlij= 0 for allj > Uisupper triangular. This means that all entries below the maindiagonal are zero. In notation,U= (uij) withuij= 0 for allj < M=LUis called is a useful trick for many computational reasons. It is much easierto compute the inverse of an upper or lower triangular matrix. Since inversesare useful for solving linear systems, this makes solving any linear systemassociated to the matrix much faster as well.
1 to be the matrix obtained by zeroing out the rst column of M. Then U 1 = 0 B @ 6 18 3 0 6 0 0 3 1 1 C A. 3. Now repeat the process by zeroing the second column of U 1 below the diagonal using the second row of U 1, and putting the corresponding entries into L 1. The resulting matrices are L 2 and U 2. For our example, L 2 = 0 B @ 1 0 0 1 3 1 ...
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