© Dennis L. Bricker Dept of Mechanical & Industrial Engineering University of Iowa Solving Linear Eqns 8/26/2003 page 1 Solving Linear Eqns 8/26/2003 page 3 Contents • Preliminaries o Elementary Matrix Operations o Elementary Matrices o Echelon Matrices o Rank of Matrices • Elimination Methods o Gauss Elimination o Gauss-Jordan Elimination • Factorization Methods o Product Form of Inverse (PFI) o LU Factorization o LDLT Factorization o Cholesky LLT Factorization Solving Linear Eqns 8/26/2003 page 2 Solving Linear Eqns 8/26/2003 page 4 Solving Linear Eqns 8/26/2003 page 5 Solving Linear Eqns 8/26/2003 page 7 Solving Linear Eqns 8/26/2003 page 6 Solving Linear Eqns 8/26/2003 page 8 Solving Linear Eqns 8/26/2003 page 9 Solving Linear Eqns 8/26/2003 page 11 Solving Linear Eqns 8/26/2003 page 10 Solving Linear Eqns 8/26/2003 page 12 Solving Linear Eqns 8/26/2003 page 13 Solving Linear Eqns 8/26/2003 page 15 Solving Linear Eqns 8/26/2003 page 14 Solving Linear Eqns 8/26/2003 page 16 Solving Linear Eqns 8/26/2003 page 17 Solving Linear Eqns 8/26/2003 page 19 Solving Linear Eqns 8/26/2003 page 18 Solving Linear Eqns 8/26/2003 page 20 Solving Linear Eqns 8/26/2003 page 21 Solving Linear Eqns 8/26/2003 page 23 Solving Linear Eqns 8/26/2003 page 22 Solving Linear Eqns 8/26/2003 page 24 Solving Linear Eqns 8/26/2003 page 25 Solving Linear Eqns 8/26/2003 page 27 Solving Linear Eqns 8/26/2003 page 26 Solving Linear Eqns 8/26/2003 page 28 Solving Linear Eqns 8/26/2003 page 29 Solving Linear Eqns 8/26/2003 page 31 Solving Linear Eqns 8/26/2003 page 30 Solving Linear Eqns 8/26/2003 page 32 Solving Linear Eqns 8/26/2003 page 33 Solving Linear Eqns 8/26/2003 page 35 Solving Linear Eqns 8/26/2003 page 34 Solving Linear Eqns 8/26/2003 page 36 CHOLESKY FACTORIZATION Cholesky factorization… Suppose that A is a symmetric & positive definite matrix. Then the Cholesky factorization of A is A = LLˆˆT where Lˆ is a lower triangular matrix. Computation: Suppose that we have the factorization A = LDLT i ≥ ˆ Then if Di 0, we can define a new diagonal matrix D where ˆ ii≡ DiiD T Then A ==L D LTT L Dˆˆ D L =()() LD ˆ LD ˆ = L ˆˆ L Twhere Lˆˆ= LD Solving Linear Eqns 8/26/2003 page 37 Solving Linear Eqns 8/26/2003 page 39 Example: The lower triangular matrix L is found by performing (on the identity matrix) the inverse of the row operations used to reduce We wish to find the Cholesky factorization of the matrix the A matrix: 201 A = 011 100 112 ←+1 RR33 R 1 2 ⇒ L = 010 RRR←+ 332 1 11 2 We now have the LU factorization of matrix A: 100201 ALU==010011 111100 22 Solving Linear Eqns 8/26/2003 page 38 Solving Linear Eqns 8/26/2003 page 40 Define the diagonal matrix D: ˆ ˆ ii≡ Define the diagonal matrix D where DiiD : 1 20 0 00 2 −1 20 0 DD= 01 0⇒ = 0 10 Dˆ = 01 0 001002 2 001 Note that 2 1 10020 0 2 0 0 0020 1 2 −1 LLDˆˆ==01001 0 = 0 1 0 UDUˆ ==010011 Then compute 111111 0 0 1 00200 1 2 2 22 2 101 So the Cholesky factorization is 2 = 01 1 20 1 20 02 00 1 ALL==ˆˆT 010 011 11100 1 22 2 Solving Linear Eqns 8/26/2003 page 41 Solving Linear Eqns 8/26/2003 page 43 And so, 1 10020010 2 ALDL==T 010010011 110011100 22 Solving Linear Eqns 8/26/2003 page 42.