LU Factorization

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

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

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

LUX = B LY = B

UX = Y

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

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

Permutation matrix: Scaling matrix: Row combination:

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

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.

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

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

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

Inverses of the three elementary matrices are:

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

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

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