Elementary Matrices and The LU Factorization

Transcription

1 lementary Matrices and The LU Factorization Definition: ny matrix obtained by performing a single elementary row operation (RO) 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. dd times row i to row j Corresponding to the three RO, we have then three elementary matrices: Type : P ij - permute rows i and j in I n. Type : M i ( ) - multiply row i of I n by a non-zero scalar Type : ij ( ) - dd times row i of I n to row j ll three types of elementary matrices are shown below: Permutation matrix: P Scaling matrix: M ( ) Row combination: ( ) Pre-multiplying an n p matrix by an n n elementary matrix has the effect of performing the corresponding RO on. xample: We can multiply the First row of the matrix by (an elementary row operation). The resulting matrix will become 6

2 We can achieve the same result by pre-multiplying by the corresponding elementary matrix. M 6 ( ) n RO can be performed on a matrix by pre-multiplying the matrix by a corresponding elementary matrix. Therefore, we can show that any matrix can be reduced to a row echelon form (RF) by multiplication by a sequence of elementary matrices. Therefore, we can write m R () where R denotes an RF of. Consider a nonsingular n n matrix. Since the unique reduced row echelon form (RRF) of such a matrix is the identity matrix I n, it follows that there exists elementary matrices (i.e. there exists elementary row operations),,..., such that m I n () But we now that I n and this implies from qn. () that equivalently m I n This shows that reduces to the identity matrix. This is what we do to find method. () m can be obtained by applying to I n the same sequence of RO that using the Gauss-Jordan LU decomposition of a nonsingular matrix nonsingular matrix can be reduced to an upper triangular matrix using elementary row operations of Type only. The elementary matrices corresponding to Type ROs are unit lower triangular matrices. We can write U () where,,..., are unit lower triangular Type elementary matrices and U is an upper triangular matrix. Since each elementary matrix is nonsingular (meaning their inverse exist) we can write from qn. () that m U () We now that the product of two lower triangular matrices is also a lower triangular matrix. Therefore qn. () can be written as LU where L (6) Of course we need to now the inverses of the Type elementary matrices. Inverses of the three n n elementary matrices are: M M P P and ( ) ( ) ( ) ( ) i i, ij ij ij ij or

3 xample: Determine the LU factorization of the matrix First, let us do the ROs to reduce into an upper triangular matrix in the following manner. ( ), ( ) ( ) These ROs can be written in terms of their equivalent elementary matrices as where () ( ), ( ), ( ) Note the order of multiplication in qn. (). U and L We can compute the inverses of the elementary matrices very easily. ( ), ( ), ( ) Therefore, L L Therefore can be written as

4 We can construct the lower triangular matrix L without multiplying the elementary matrices if we utilize the multipliers obtained while we converted matrix into an upper triangular matrix. But, what exactly are those multipliers? Definition: When using RO of Type, 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 ji m. In our example we have three multipliers: m, m, m If we notice the unit lower triangular matrix L carefully, we see that the elements beneath the leading diagonal are just the corresponding multipliers. This relationship holds in general. Therefore, we can do elementary row operations of Type to reduce to upper triangular form and then utilize the corresponding multipliers to write L directly. xample: Determine the LU factors for the matrix 6 : Type ROs to reduce to the upper triangular matrix can be achieved by premultiplying by the corresponding elementary matrices. The elementary matrices are listed in the order they are multiplied. : : :

5 : : : If you lie the fractional form then U 8 : The lower triangular matrix L can be found by the following. L : L If you lie the fractional form then

6 L : Note that the multipliers corresponding to the ROs are: m : m : m : m : m : m : In a unit lower triangular case, the matrix L can be constructed directly by utilizing the multipliers. To verify multiply L and U. LU 6 TH ND