Gram-Schmidt Orthogonalization

Presentation on theme: "Gram-Schmidt Orthogonalization"— Presentation transcript:

1 Gram-Schmidt Orthogonalization
MA2213 Review Lectures 1-4 Inner Products Gramm Matrices Gram-Schmidt Orthogonalization

2 Transpose and its Properties
Theorem 1 and is positive definite Proofs

3 Inner ( = Scalar) Product Spaces
is a vector space over reals with an inner product that satisfies the following 3 properties: symmetry linearity positivity Remark Symmetry and Linearity imply hence (- , -) : V x V R is Bilinear

4 Examples of Inner Product Spaces
positive definite, symmetric Remark The standard inner product on is obtained by choosing then Example 2. ( is called a weight function) and Remark The SIP on is obtained by choosing

5 The Gramm Matrix of a finite sequence of vectors in an inner product space V is the matrix Theorem 2 Let are the columns vectors of Then is the Gramm matrix of the sequence Proof

6 Standard Basis Definition The standard (sequence) of basis vectors for is where

7 Questions Question 1. What is the following matrix Question 2. What is the following ? if Question 2. For the standard inner product on what is ?

8 Gram-Schmidt Orthogonalization
Theorem 3. Given a sequence of linearly independent vectors in an inner product space there exists a unique upper triangular matrix with diagonal entries 1 such that the ‘matrix’ has orthogonal column vectors. Proof Since it suffices to show that for these are n-1 systems with Gramm matrices

9 Gram-Schmidt Algorithm
start with

10 Gram-Schmidt Orthonormalization
produce an orthonormal basis start with Here