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 > = ( Sx ) T Sy . 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) < cx, y > = c < x, y >, c R (iv) < x, x > > 0 , x R n such that x = 0 Let’s present < x, y > in the following form < x, y > = ( Sx ) T Sy = x T S T Sy . If we denote S T S by B (notice that B is symmetric) then < x, y > = x T By x, y R n . Let’s check (i)-(iv). (i) < x, y > = x T By = ( x T By ) T = y T Bx = < y, x > (ii) < x 1 + x 2 , y > = ( x 1 + x 2 ) T By = x T 1 By + x T 2 By = < x 1 , y > + < x 2 , y > (iii) < cx, y > = cx T By = c < x, y >, c R (iv) < x, x > = ( Sx ) T Sx = (let z = Sx ) = z T z = z 2 > 0 z R n such that z = 0 So z = Sx must be different from 0 for any x = 0. It holds iff the system of linear equations Sx = 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: For which S < x, y > = x · y (dot product). Solution: From fact 5.3.6 we know that x · y = x T y . And < x, y > = x T S T Sy = x T y S T S = I . Hence, S must be orthogonal (fact 5.3.8).
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