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.
Using LU Decomposition to Solve Linear Systems Suppose we have M= LUand want to solve the system MX= LUX= V: Step 1: Set W= 0 B @ u v w 1 C A= UX. Step 2: Solve the system LW= V. This should be simple by forward substitution since Lis lower triangular. Suppose the solution to LW= V is W 0. Step 3: Now solve the system UX = W 0. This should be ...
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