Scientific Computing

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CS205 Review Session #6 Notes More Norms Let A be an n × n positive definite matrix. We can write A as A = MM T where M is an appropriate n × n matrix. There are many choices for M . For example, using Cholesky factorization, we can write A = LL T where L is a lower triangular matrix. Alternatively, using the diagonal form of A we can write: A = QΛQ T = 1 / 2 Λ 1 / 2 Q = ( 1 / 2 )( 1 / 2 ) T Using any such matrix M allows us to express u , v A = u T Av = u T MM T v = M T u · M T v Therefore, the inner product induced by A is equivalent to transforming our vector space into a new vector space via the mapping x M T x , and then taking the usual Euclidean dot product into the transformed space. This can be used to prove that the norm x A = xA T x = M T 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 { x 1 , x 2 , . . . , x k } be a set of vectors in IR n . The Gram-Schmidt algorithm for creating an
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