Contents • Preliminaries O Elementary Matrix Operations O

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

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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

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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 0020 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 02 00 1  ALL==ˆˆT 010 011  11100 1 22 2

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And so, 1 10020010 2 ALDL==T 010010011  110011100 22

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