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LU Factorization

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Equating the elements of the First Row :- Equating the elements of the 2nd Row :- Equating the elements of the 3rd Row :-

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We have 12 unknowns but only 9 equations. We need some sort of compromise. Crout’s Method Set Dolittle’s Method Set

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Solve for y, and then solve for x. Use of LU factors in solving systems of linear equations

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LUX = B LY = B

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UX = Y

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Elementary Matrices and The LU Factorization Definition: Any matrix obtained by performing a single elementary row operation (ERO) on the identity (unit) matrix is called an elementary matrix. There are three elementary operations: Permute rows i and j Multiply row i by a non-zero scalar k Add k times row i to row j

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Corresponding to the three ERO, we have then three elementary matrices: Type 1: - permute rows i and j in In. Type 2: - multiply row i of In by a non-zero scalar k Type 3: - Add k times row i of In to row j

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Permutation matrix: Scaling matrix: Row combination:

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Pre-multiplying a matrix A by an elementary matrix E has the effect of performing the corresponding ERO on A. Example: We can multiply the First row of the matrix A by 3 (an elementary row operation). The resulting matrix will become

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We can achieve the same result by pre-multiplying A by the corresponding elementary matrix. An ERO can be performed on a matrix by pre- multiplying the matrix by a corresponding elementary matrix. Therefore, we can show that any matrix A can be reduced to a row echelon form (REF) by multiplication by a sequence of elementary matrices.

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where R denotes an REF of A. Since the unique reduced row echelon form (RREF) of a matrix is the unit matrix

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A nonsingular matrix can be reduced to an upper triangular matrix using elementary row operations of Type 3 only. The elementary matrices corresponding to Type 3 EROs are unit lower triangular matrices. We can write

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Since each elementary matrix is nonsingular (meaning their inverse exist) we can write We know that the product of two lower triangular matrices is also a lower triangular matrix. Therefore

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Inverses of the three elementary matrices are:

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Determine the LU factorization of the matrix First, let us do the EROs to reduce A into an upper triangular matrix.

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These EROs can be written in terms of their equivalent elementary matrices as

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We can construct the lower triangular matrix L without multiplying the elementary matrices if we utilize the multipliers obtained while we converted matrix A into an upper triangular matrix. Definition: When using ERO of Type 3, the multiple of a specific row i that is subtracted from row j to put a zero in the ji position is called a multiplier, and is denoted as

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