Section 9 Notes - MATH 294 Chapter 5.5 Inner Product Spaces...

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MATH 294 Chapter 5.5 Inner Product Spaces Ex 5.5.3 Let S be a n × n matrix. In R n define the product < ~x,~ y > = ( S~x ) T S~ y . Part a: For which choices of S is this an inner product? Solution: We know that an inner product must fulfill the following properties (definition 5.5.1): (i) < ~x,~ y > = < ~ y,~x > (ii) < ~x 1 + ~x 2 ,~ y > = < ~x 1 ,~ y > + < ~x 2 ,~ y > (iii) < c~x,~ y > = c < ~x,~ y >, c R (iv) < ~x,~x > > 0 , ~x R n such that ~x 6 = ~ 0 Let’s present < ~x,~ y > in the following form < ~x,~ y > = ( S~x ) T S~ y = ~x T S T S~ y . If we denote S T S by B (notice that B is symmetric) then < ~x,~ y > = ~x T B~ y ~x,~ y R n . Let’s check (i)-(iv). (i) < ~x,~ y > = ~x T B~ y = ( ~x T B~ y ) T = ~ y T B~x = < ~ y,~x > (ii) < ~x 1 + ~x 2 ,~ y > = ( ~x 1 + ~x 2 ) T B~ y = ~x T 1 B~ y + ~x T 2 B~ y = < ~x 1 ,~ y > + < ~x 2 ,~ y > (iii) < c~x,~ y > = c~x T B~ y = c < ~x,~ y >, c R (iv) < ~x,~x > = ( S~x ) T S~x = (let ~ z = S~x ) = ~ z T ~ z = k ~ z k 2 > 0 ~ z R n such that ~ z 6 = ~ 0 So ~ z = S~x must be different from ~ 0 for any ~x 6 = ~ 0. It holds iff the system of linear equations S~x = ~ 0 has an unique solution ~x = ~ 0. So we’re going to get (iv) holds S is invertible. So < ~x,~ y > is inner product S is invertible. Part b:
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This note was uploaded on 09/26/2007 for the course MATH 2940 taught by Professor Hui during the Fall '05 term at Cornell University (Engineering School).

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Section 9 Notes - MATH 294 Chapter 5.5 Inner Product Spaces...

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