Ee1034dS10

Ee1034dS10 - Choleski Factorization Ax b m n rankA n Ex 1...

This preview shows pages 1–6. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Choleski Factorization , , Ax b m n rankA n Ex 1: Linear Least Squares , , Ax b m n rankA n e x A x b ( ) 2 || ( ) || ( ) ( ) ( ) ( ) t t e x e x e x Ax b Ax b 2 || ( ) || ( ) ( ) ( ) ( ) ( ) ( ) ( ) t e x e x e x Ax b Ax b f x Ax b Ax b 2 2 min || ( ) || min ( ) min( ) ( ) t x x x e x f x Ax b Ax b Expanding the term on the right hand side, we have ( ) 2 t t t t f x x A Ax b Ax b b EE103 SLIDES 4D (SEJ) 1 Example: LLS ( ) 2 2 t t t f x x A A b A or 1 ) ) ( ( t t not for computa x A A A b tion ( ) t t t f x A Ax A b ) ) ( ( not for computa x A A A b tion b 1 ( ) def t t A x A A A A b b a1 b a2 EE103 SLIDES 4D (SEJ) 2 PD Matrices: Choleski Factorization x A x x T x Ax a x x a x T ii i i n i ij j j n i n 2 1 1 1 i j j i i 1 1 1 If A is PD then A is nonsingular (the columns are linearly independent and, of course, so are the rows). t Let x and assume Ax A x x A x EE103 SLIDES 4D (SEJ) 3 PD Matrices: Choleski Factorization i i Also a for all i , , 1 1 1 0 i i T To seethat a assumethat a Let x Then , , , ( , , , ) 1 1 T x Ax a Thiscontradictstheassumptionthat Ais PD , Thiscontradictstheassumptionthat Ais PD EE103 SLIDES 4D (SEJ) 4 PD Matrices: Choleski Factorization If A is PD then so is its invers x If A is PD then so is its inverse First note that the transpose is also PD T T T T T x x Ax x Ax x A x Now let x A y 1 x y A x 1 1 T T Now let x A y x A x Ay A Ay , 1 ( P D ) T T T T T y A A Ay y A y y Ay EE103 SLIDES 4D (SEJ) 5 0 (PD) y A y y Ay 1 T x A x Choleski Factorization ? T A symmetric A LL , If this can be done with 0, 1, , i i l i n then A is PD T T T T x x Ax x LL x y y EE103 SLIDES 4D (SEJ) 6 PD Matrices: Choleski Factorization Let A be an m x n matrix, m n and assume the columns of A are linearly independent. Then A A T is symmetric and positive definite. 1: ( ) T T T A A A A 2: 0. def T T T Let x Then x A Ax y y y A x EE103 SLIDES 4D (SEJ) 7 Example: Unconstrained Minimization min ( , , ) min ( ) n f x x x f x 1 2 Ex 2: Unconstrained Minimization , , , x x x n x n 1 2 1 2 EE103 SLIDES 4D (SEJ) 8 i i f Example: Unconstrained Minimization...
View Full Document

{[ snackBarMessage ]}

Page1 / 11

Ee1034dS10 - Choleski Factorization Ax b m n rankA n Ex 1...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online