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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 GramSchmidt algorithm for creating an orthogonal set {~1 , x2 , . . . , xk } is given by the recurrence: x ~ ~
i1 xi = xi  ~
j=1 xi xj ~ xj ~ xj xj ~ ~ The corresponding algorithm for creating a set of Aorthogonal vectors is
i1 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 , . . . , xi1 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 ...
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

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