LU - Factorizations Matrix Factorization into Triangular Matrices

LU Factorization The LU Factorization of a n  n square Matrix M = (m ij ) is a way of expressing the Matrix M as a product of two square n  n matrices L and U where the matrix L is a lower triangular matrix and The matrix U is an upper triangular matrix. Furthermore the matrix L has all 1's on the diagonal. Triangular Matrices A n  n square matrix L =( l ij ) is called Lower Triangular if l ij = 0 if i < j (i.e. all entries above the diagonal are zero). A n  n square matrix U =( u ij ) is called Upper Triangular if u ij = 0 if i > j (i.e. all entries below the diagonal are zero). Lower Triangular Matrix Upper Triangular Matrix

LU Factorization Method The factorization is done by keeping tract of a series of elementary matrices where multiplication on the left by an elementary matrix corresponds to a row operation. 1. Apply the necessary row operations to get the matrix to be upper triangular. 2. For each row operation multiply on the right by an elementary matrix to get U. 3. Multiply on the right by the inverse of all of the elementary matrices to get L.

Elementary Matrices The elementary matrices E k must be of a certain type that correspond to multiplying a row by a number and adding it to another row. R j = cR i + R j j th row Consider the row operation of multiplying row i by c and adding it to row j ( i ≠ j ). This row operation is abbreviated by R j = cR i + R j. The corresponding elementary matrix has 1's going down the diagonal, a c in row j and column i, and zeros in every other position. i th column 1. The inverse of a matrix like this just puts a –c in the position where c is. 2. The product of two of these type of matrices with a c 1 and c 2 in two different positions results in a matrix with c 1 in the position where it was and c 2 in the position it was in and 1's on the diagonal and zeros everywhere else. 3. If all row operations have i < j the matrix will be lower triangular.

LU Factorization Method (Example) The following is an example of how to factor a 3  3 matrix and keep tract of the elementary row reduction matrices. R 2 = -2 R 1 + R 2 R 3 = 3 R 1 + R 3 The matrix is now upper triangular we combine the two multiplications. Use matrix arithmetic and substitution.

Calculating the inverse for elementary matrices of this type can be done quickly (i.e. a shortcut) by just taking the negative of the nonzero, non- diagonal entry. Multiply by the inverses from previous slide. Substitute and group elementary matrices A short cut for multiplying these matrices is putting the entries in the corresponding positions.

LU Factorization (Short Cut) We do not need to do all the moving around of matrices on each side of the equation. We just keep tract of the row operations and fill in the entries of the corresponding matrix to get a sequence of lower triangular matrices. R 2 =-4 R 1 + R 2 R 3 =5 R 2 + R 3 Stop here the matrix is upper triangular and this is the matrix U. Can check that the factorization is correct with a matrix multiplication

Matrices Without LU Factorizations Not every matrix has an LU Factorization, just like not all numbers or polynomials can be factored. Look at the simple 2  2 matrix M given to the right. To show this is impossible to factor this way consider what L and U would need to do. L is lower triangular and U is upper triangular. M = LU This means that ae = 0 so that either a = 0 or e =0 If a =0 then detL =0 so the detLU =0 If e =0 then detU =0 so the detLU =0 When row reducing the matrix if you ever need to do the row operation of interchanging (swapping) two rows the matrix can not be LU factored.