Ee1034dS10

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

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

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