lecture6 - CSE/MATH 6643: Numerical Linear Algebra Haesun...

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Unformatted text preview: CSE/MATH 6643: Numerical Linear Algebra Haesun Park { hpark } @cc.gatech.edu School of Computational Science and Engineering College of Computing Georgia Institute of Technology Atlanta, GA 30332, USA Lecture 6 Symmetric Positive Definite & Cholesky Decomp. 5 Symmetric Positive Definite & Cholesky Decomp. 5.1 Symmetric Positive Definite Matrix A : symmetric positive definite (S.P.D) A T = A and x T Ax > for any x negationslash = 0 . e.g. A = " 5 1 1 4 # CSE/MATH 6643: Numerical Linear Algebra p.1/10 Symmetric Positive Definite & Cholesky Decomp. Lemma. In a S.P.D. matrix, max 1 i n, 1 j n | a ij | lies on diagonal and all diagonal elements are positive. I n = h e 1 e n i . Since A is symmetric positive definite, a ii = e T i Ae i > for any i = 1 , ,n . So a ii > ( e i + e j ) T A ( e i + e j ) = a ii + a jj + 2 a ij > ( e i e j ) T A ( e i e j ) = a ii + a jj 2 a ij > So we have | a ij | a ii + a jj 2 max { a ii ,a jj } CSE/MATH 6643: Numerical Linear Algebra p.2/10 Symmetric Positive Definite & Cholesky Decomp.Symmetric Positive Definite & Cholesky Decomp....
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This note was uploaded on 02/04/2012 for the course CS 8801 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

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lecture6 - CSE/MATH 6643: Numerical Linear Algebra Haesun...

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