This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: EE103 (Fall 200910) 7. The condition number • Ax = b when A is singular • condition of a set of linear equations • matrix norm • condition number 71 Linear equations with singular coefficient matrix if A is nonsingular, then Ax = b has a unique solution for every b if A is singular, then Ax = b has zero or infinitely many solutions: • for some righthand sides b , the equation Ax = b has no solution for example, we can take a nonzero vector b in the nullspace of A T Ax = b = ⇒ 0 = b T Ax = b T b > hence Ax = b is unsolvable • if Ax = b is solvable, then it has infinitely many solutions: A ( x + tv ) = b for all scalar t , and all nonzero vectors v in the nullspace of A The condition number 72 Example A = bracketleftbigg 1 − 1 − 2 2 bracketrightbigg is singular: Ax = 0 for x = (1 , 1) • the equation Ax = b has no solution for b = (1 , 0) • the equation Ax = b has infinitely many solutions for b = (2 , − 4) : bracketleftbigg x 1 x 2 bracketrightbigg = bracketleftbigg 2 + t t bracketrightbigg is a solution, for any t The condition number 73 Condition of a set of linear equations assume A is nonsingular and Ax = b if we change b to b + Δ b , the new solution is x + Δ x with A ( x + Δ x ) = b + Δ b the change in x is Δ x = A − 1 Δ b ‘condition’ of the equations: a technical term used to describe how sensitive the solution is to changes in the righthand side...
View
Full Document
 Spring '08
 VANDENBERGHE,LIEVEN
 Linear Algebra, Numerical Analysis, Matrices, Singular value decomposition, condition number

Click to edit the document details