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
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
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