chol - EE103(Fall 2009-10 5 The Cholesky factorization •...

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Unformatted text preview: EE103 (Fall 2009-10) 5. The Cholesky factorization • positive (semi-)definite matrices • examples • the Cholesky factorization • solving Ax = b with A positive definite • inverse of a positive definite matrix • permutation matrices • sparse Cholesky factorization 5-1 Positive (semi-)definite matrices • A is positive definite if A is symmetric and x T Ax > for all x negationslash = 0 • A is positive semidefinite if A is symmetric and x T Ax ≥ for all x Note: if A is symmetric of order n , then x T Ax = n summationdisplay i =1 n summationdisplay j =1 a ij x i x j = n summationdisplay i =1 a ii x 2 i + 2 summationdisplay i>j a ij x i x j The Cholesky factorization 5-2 Examples A 1 = bracketleftbigg 9 6 6 5 bracketrightbigg , A 2 = bracketleftbigg 9 6 6 4 bracketrightbigg , A 3 = bracketleftbigg 9 6 6 3 bracketrightbigg • A 1 is positive definite: x T A 1 x = 9 x 2 1 + 12 x 1 x 2 + 5 x 2 2 = (3 x 1 + 2 x 2 ) 2 + x 2 2 • A 2 is positive semidefinite but not positive definite: x T A 2 x = 9 x 2 1 + 12 x 1 x 2 + 4 x 2 2 = (3 x 1 + 2 x 2 ) 2 • A 3 is not positive semidefinite: x T A 3 x = 9 x 2 1 + 12 x 1 x 2 + 3 x 2 2 = (3 x 1 + 2 x 2 ) 2 − x 2 2 The Cholesky factorization 5-3 Examples • A = B T B for some matrix B x T Ax = x T B T Bx = bardbl Bx bardbl 2 A is positive semidefinite A is positive definite if B has a zero nullspace • diagonal A x T Ax = a 11 x 2 1 + a 22 x 2 2 + ··· + a nn x 2 n A is positive semidefinite if its diagonal elements are nonnegative A is positive definite if its diagonal elements are positive The Cholesky factorization 5-4 Another example A = 1 − 1 ··· − 1 2 ··· . . . . . . . . . . . . . . . ··· 2 − 1 ··· − 1 1 A is positive semidefinite: x T Ax = ( x 1 − x 2 ) 2 + ( x 2 − x 3 ) 2 + ··· + ( x n- 1 − x n ) 2 ≥ A is not positive definite: x T Ax = 0 for x = (1 , 1 ,..., 1) The Cholesky factorization 5-5 Resistor circuit y 1 y 2 x 1 x 2 R 1 R 2 R 3 Circuit model: y = Ax with A = bracketleftbigg R 1 + R 3 R 3 R 3 R 2 + R 3 bracketrightbigg ( R 1 ,R 2 ,R 3 > 0) Interpretation of x T Ax = y T x x T Ax is the power delivered by the sources, dissipated by the resistors The Cholesky factorization 5-6 A...
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chol - EE103(Fall 2009-10 5 The Cholesky factorization •...

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