LU FactorizationIn Matlab there are several built-in functions provided for matrix factorization (also called decomposition). The name of the built-in function for a Lower-Upper decomposition is 'lu'. To get the LU factorization of a square matrix A, type the command '[L, U] = lu(A)'. Matlab returns a lower triangular matrix L and an upper triangular matrix U such that L*U = A. Suppose that A= [ 1 2 -3 -3 -4 13 2 1 -5] We can verify that if L = [ 1 0 0 -3 1 0 2 -1.5 1] and U = [1 2 -3 0 2 4 0 0 7] Then, A = L*U The decomposition of the matrix A is an illustration of an important and well known theorem. If A is a nonsingular matrix that can be transformed into an upper diagonal form U by the application or row addition operations, then there exists a lower triangular matrix L such that A = LU. Row addition operations can be represented by a product of elementary matrices. If n such operations are required, the matrix U is related to the matrix A in the following way: U = En
En-1
... E2
E1
A L = E1-1
E2-1
... En-1 is equal to the multiplier used to eliminate the (i, j) position. Example: In Matlab, let's find the LU decomposition of the matrix A = [-2 1 -3; 6 -1 8; 8 3 -7] Write this instruction in the command window or within a script: [L, U] = lu(A) And the Matlab answer is: L = -0.2500 -0.5385 1.0000 0.7500 1.0000 0 1.0000 0 0 U = 8.0000 3.0000 -7.0000 0 -3.2500 13.2500 0 0 2.3846 We can test the answer, by typing L*U And, finnally, the Matlab answer is: ans = -2.0000 1.0000 -3.0000 6.0000 -1.0000 8.0000 8.0000 3.0000 -7.0000 >> Showing that A = L*U, indeed. From 'LU Factorization' to home From 'LU Factorization' to 'Linear Algebra Menu'
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