CS205A review_6

Scientific Computing

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CS205 Review Session #6 Notes More Norms Let A be an n n positive definite matrix. We can write A as A = MMT where M is an appropriate n n matrix. There are many choices for M. For example, using Cholesky factorization, we can write A = LLT where L is a lower triangular matrix. Alternatively, using the diagonal form of A we can write: A = QQT = Q1/2 1/2 Q = (Q1/2 )(Q1/2 )T Using any such matrix M allows us to express u, v A = uT Av = uT MMT v = MT u MT v Therefore, the inner product induced by A is equivalent to transforming our vector space into a new vector space via the mapping x MT x, and then taking the usual Euclidean dot product into the transformed space. This can be used to prove that the norm x A = xAT x = MT x 2 satisfies the properties of a norm. In a similar way, we can show that the inner product induced by A has the properties of a regular dot product. Conjugate Vectors Let {x1 , x2 , . . . , xk } be a set of vectors in IRn . The Gram-Schmidt algorithm for creating an orthogonal set {~1 , x2 , . . . , xk } is given by the recurrence: x ~ ~ i-1 xi = xi - ~ j=1 xi xj ~ xj ~ xj xj ~ ~ The corresponding algorithm for creating a set of A-orthogonal vectors is i-1 xi = xi - ~ j=1 xi A~j x xj ~ xj A~j ~ x Note that in the computation of xi we subtract a linear combination of x1 , x2 , . . . , xi-1 from ~ ~ ~ ~ xi . Problem 3.5 For an n n matrix A we have |xT Ax| = |x Ax| x 2 A = max x 2 2 Ax 2 x 2 A 2 x 2 1 Furthermore, if A is symmetric we have A = QQT and xT Ax xT QQT x = T xT x x QQT x This allows us to observe that: min xT Ax max xT x y=QT x = yT y = yT y 2 i y i 2 yi 2 ...
View Full Document

This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

Page1 / 2

CS205A review_6 - CS205 Review Session#6 Notes More Norms Let A be an n n positive definite matrix We can write A as A = MMT where M is an

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online