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Unformatted text preview: NOTES ON SECTION 5 . 3 MA265, SECTIONS 41, 52 Some key words Inner Product Inner Product Space Symmetric Matrix Positive Definite Matrix Distance Angle Cauchy-Shwarz inequality Triangle Inequality Orthogonal Orthonormal 1. Pairing We are now going to abstract the dot product for R 2 , R 3 to other vector spaces (how to do so is obvious for R n , but there are also important and useful generalizations to arbitrary vector spaces which requires some more thought as to how we should set it up). Suppose V is a vector space. We suppose that we have a method to assign to two vectors u , v V a number ( u , v ) R which we call a pairing. If we dont say anything about this pairing its not going to be very useful, so we are going to put some demands on it now. Example 1. We can define a pairing ( x , y ) on R n by taking any A M n,n ( R ) and defining ( x , y ) = x T A y R The usual dot product is obtained from the choice A = I . 2. Inner Product Spaces We turn the Theorem from the last section (5.1) into a definition: Definition 2. Suppose the pairing ( u , v ) satisfies the following identities: (a) ( u , u ) , and is equal to zero if and only if u = . (b) ( u , v ) = ( v , u ) (c) (( u + v ) , w )) = ( u , w ) + ( v , w ) (d) ( c u , v ) = c ( u , v ) Then we say ( , ) is a inner product . We call ( V, ( , )) an inner product space . Example 3. R 2 , R 3 with the dot product forms an inner product space. So does R n with its dot product. But there are many many others, even on R n . To start off with, we need a pairing. So lets take the pairing of example 1. However, this pairing will almost never be an inner product if we dont impose some hypotheses on A . Definition 4. A symmetric matrix A is called positive definite if for every non-zero vector u 6 = u T Au > As an example, the identity matrix is positive definite, because u T I u = u T u = n X i =1 u 2 i > 0 if u 6 = It might not be immediately obvious that there are other positive definite matrices, but in fact there are many and well give some examples later. First, we show how they give inner products Example 5. (Theorem 5.2 in textbook) The pairing associated to a positive definite symmetric matrix A = A T is an inner product, that is ( u , v ) = u T A v 1 Property ( a ) follows precisely from the assumption that A is positive definite (and its definition). The proof of property ( b ) uses the fact that A is symmetric (its a two line proof but see the textbook because its important!) Properties ( c ) , ( d ) are almost immediate from basic matrix algebra (section 1.4, Theorems 1 . 2, etc...)....
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