v 1 2 v 1 is orthogonal to v 1.

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1 Gram-Schmidt Process This process consists of steps that describes how to obtain an orthonormal basis for any finite dimensional inner products. Let V be any nonzero finite dimensional inner product space and suppose that {u 1, u,..., u n } is any basis for V. We will form an orthogonal basis from this basis say { 1,,..., n } Step 1: Let 1 = u 1 Step : Produce a ector that is orthogonal to 1. Hence = u proj 1 u = u <u, 1 > 1 1 is orthogonal to 1. Step : Produce a ector that is orthogonal to both 1 and, equialently, a ector that is orthogonal to the space W spanned by 1 and. Hence = u proj W u = u <u, 1 > 1 1 <u, > is orthogonal to both 1 and. We will continue this way untill we produce n ectors that is n = u n proj Wn 1 u n = u n < u n, 1 > 1 1 < u n, >... < u n, n 1 > n 1 Then { 1,,..., n } will be an orthogonal basis of V and { 1 will be an orthonormal basis of V. 1,,..., n n 1 n } Example: Let R hae the Euclidean inner product. Use Gram-Schmidt process to transform the basis {(1, 1, 1), ( 1, 1, 0), (1,, 1)} into an orthonormal basis. 1

2 Solution: Let 1 = (1, 1, 1), then = ( 1, 1, 0) proj 1 ( 1, 1, 0) = ( 1, 1, 0) and = (1,, 1) proj 1 (1,, 1) proj (1,, 1) = ( 1, 1, 1) =, = 1, = {( 1 1 6, 1, ), ( 1 1,, 0), ( 1 1 6, 6, 6)} is an orthonormal basis. Example: Consider the ector space M of real matrices with an[ inner] product [ ] defined [ ] as < A, B >= tr(a T B) where A and B are matrices. Let S={,, } be a basis of a subspace W of inner product space M. Find an orthogonal basis for W by Gram-Schmidt process. Solution: [ ] = 0 0 [ ] [ ] [ ] [ ] = proj ( ) = 0 = [ ] [ ] [ ] [ ] [ ] [ ] = proj ( ) proj 1 1 ( ) = 0 = [ ] [ ] [ ] Hence {,, } is an orthogonal basis for W Theorem 6..6 If W is a finite dimensional inner product space, then: (a) Eery orthogonal set of nonzero ectors in W can be enlarged to an orthogonal basis for W. (b) Eery orthonormal set in W can be enlarged to an orthonormal basis for W. Example: Show that {(1,, ), (4, 1, )} is an orthogonal set in R and extend it to be an orthogonal basis for R by adding appropriate ector to it. Solution: (1,, ) (4, 1, ) = 0 hence they are orthogonal. Find = (a, b, c) which is not in the span{(1,, ), (4, 1, )}. That is find a,b,c such that the system below is inconsistent c 1 + 4c = a c 1 + c = b c 1 c = c

3 1 4 a Row echelon form of the augmented matrix is a b 0 1. Hence for a + c b a + c b will be outside of the span{(1,, ), (4, 1, )}. That is {(1,, ), (4, 1, ), (1,, 0)} forms a basis for R. Now apply Gram-Schmidt to form an orthogonal basis. 1 = (1,, ) = (4, 1, ) proj (1,,) (4, 1, ) = (4, 1, ) = (1,, 0) proj (1,,) (1,, 0) proj (4,1, ) (1,, 0) = ( 5 6, 5, 5 6 ) Hence {(1,, ), (4, 1, ), ( 5 6, 5, 5 6 )} is an orthogonal basis for R. Chapter 7 Diagonalization and Quadratic Forms Section 7.1 Orthogonal Matrices Definition: A square matrix A is said to be orthogonal if its transpose is the same as its inerse, that is if A 1 = A T or equialently, if AA T = A T A = I [ Example: Standart ] matrix for counterclockwise rotation in R through an angle θ cos (θ) sin(θ) = is orthogonal. sin (θ) cos θ [ ] T [ ] cos (θ) sin(θ) cos (θ) sin(θ) = sin (θ) cos θ sin (θ) cos θ [ ] Theorem 7.1.1: The following are equialent for an n n matrix A. (a) A is orthogonal (b) The row ectors of A form an orthonormal set in R n with the Euclidean inner product. (c) The column ectors of A form an orthonormal set in R n with the Euclidean inner product. Theorem 7.1. Properties: (a) The inerse of an orthogonal matrix is orthogonal (b) A product of orthogonal matrices is orthogonal (c) If A is orthogonal, then det A = 1 or det A = 1

4 Idea: (a) Let A be an orthogonal matrix then A 1 = A T. A 1 is also orthogonal as (A 1 ) 1 = A = (A 1 ) T. (b) If A and B are orthogonal matrices then A 1 = A T and B 1 = B T. Now (AB) 1 = B 1 A 1 = (B T A T ) = (AB) T. (c) Let A be an orthogonal matrix then A T A = I det (A T A) = 1 det (A T ) det (A) = 1 (det (A)) = 1 det (A) = 1 or det (A) = 1 Definition: If A is an orthogonal matrix and T A : R n R n is a matrix transformation defined by multiplication by A, then T A is called an orthogonal operator on R n. These operators sends a ector to a ector with the same length, that is they presere length of ectors. These operators preseres angles between ectors and so orthogonality. These are mentioned in the following theorem. Theorem 7.1.: If A is an n n matrix, then the following are equialent. (a) A is orthogonal (b) Ax = x for all x in R n (c) Ax Ay = x y for all x and y in R n Example: Standard matrices of rotation and reflection operations in R and R are orthogonal matrices as they presere length of the ectors under rotation. They also presere the angles between ectors. Theorem 7.1.4: If S is an orthonormal basis for an n-dimensional inner product space V, and if then (a) u = u 1 + u u n (u) S = (u 1, u,..., u n ) and () S = ( 1,..., n ) (b) d(u, ) = (u 1 1 ) + (u ) (u n n ) (c) < u, >= u u u n n Example: Consider R with the inner product < u, >= u u + u where u = (u 1, u, u ), = ( 1,, ). With this inner product S = { e 1, e, e } is an orthonormal basis for R. Then 4

5 (, 5, 6) = < (, 5, 6), (, 5, 6) > = 98 [(, 5, 6)] S = ( 6, 10, 6) By Theorem using coordinate ector of (, 5, 6) we also get (, 5, 6) = 98 Theorem 7.1.5: Let V be a finite dimensional inner product space. If P is the transition matrix from one orthonormal basis for V to another orthonormal basis for V, then P is an orthogonal matrix. Section 7. Orthogonal Diagonalization In this section we will diagonalize a square matrix by using orthogonal matrices. Definition: If A and B are square matrices, then we say that A and B are orthogonally similar if there is an orthogonal matrix P such that P T AP = B. Definition: If A is orthogonally similar to some diagonal matrix, say P T AP = D, then we say that A is orthogonally diagonalizable and that P orthogonally diagonalizes A. A way to decide if A is orthogonally diagonalizable: Definition: If A is an n n matrix, then the following are equialent (a) A is orthogonally diagonalizable (b) A has an orthonormal set of n eigenectors (c) A is symmetric Theorem 7..: If A is symmetric matrix, then: (a) The eigenalues of A are all real numbers (b) Eigenectors from different eigenspaces are orthogonal Orthogonally Diagonalizing an n n Symmetric Matrix Step1: Find a basis for each eigenspace of A Step: Apply Gram-Schmidt process to each of these bases to obtain an orthonormal basis for each eigenspace say {p 1, p,..., p n }. Step: Form the matrix P whose columns are the ectors {p 1, p,..., p n }. Then P T AP = D will be a diagonal matrix with diagonal entries as the eigenalues λ 1,..., λ corresponding to eigenectors p 1,..., p n. 5