Class Notes (810,031)
Mathematics (837)
MATB24H3 (13)
Lecture

# Lecture17-18f.pdf

12 Pages
155 Views

School
University of Toronto Scarborough
Department
Mathematics
Course
MATB24H3
Professor
Sophie Chrysostomou
Semester
Winter

Description
University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT B24S Fall 2011 MAT B24 Lectures 17-18. THEOREM: Properties of Ax for an Orthogonal Matrix ALet A be an orthogonal n × n matrix and x and y are column vectors in R . Then: 1. (Ax) ▯ (Ay) = x ▯ y Preservation of dot product. 2. ||Ax|| = ||x||Preservation of length. 3. the angle between nonzero vectors x and y equals the angle between Ax and Ay proof. T T T T T T T 1. (Ax)▯(Ay) = (Ax) Ay = (x A )Ay = x (A A)y = x Iy = x y = x▯y 2. Since (Ax) ▯ (Ax) = x ▯ x then ||Ax|| = ||x|| or ||Ax|| = ||x||. 3. If the angle between the nonzero vectors Ax and Ay equals θ then (Ax) ▯ (Ax) x ▯ x cosθ = = ||Ax|| ▯ ||Ay|| ||x||||y|| Showing that θ is also the angle between the nonzero vectors x and y. 1 THEOREM: Orthogonality of Eigenspaces of a Real Symmetric Matrix: Let A be a real symmetric matrix and λ and λ be1distinc2 eigenvalues of A. Then the eigenspaces E λ1 and E λ2 are orthogonal. proof. : To show that E λ and E λ are orthogonal we must prove v and 1 1 2 v 2re orthogonal for all v ∈ E1 λ1 and v ∈2E λ2. Let v 1 E λ1 and v ∈2E λ2. Then Av = λ v and Av = λ v . 1 1 1 2 2 2 Also, λ (v ▯v ) = (λ v )▯v = (Av )▯v = (Av ) v = (v A )v = (v A)v = v (Av ) T T 1 1 2 1 1 2 1 2 1 2 1 2 1 2 1 2 = v (λ v2) 2 λ (v 2 ) = λ2(v ▯ v2) 1 2 1 1 Thus λ (v ▯ v ) = λ (v ▯ v ) or (λ − λ )(v ▯ v ) = 0 1 1 2 2 1 2 1 2 1 2 Since λ ▯= λ then λ − λ ▯= 0 thus v ▯ v = 0 1 2 1 2 1 2 This brings us to another theorem: THEOREM: Fundamental Theorem of Real Symmetric Matrices: Every −1 real symmetric matrix A is diagonalizable. The diagonalization C AC = D can be achieved by using a real orthogonal matrix C. 2   4 2 2 EXAMPLE: Let A = 2 4 2 . Find an orthogonal matrix C that diag- 2 2 4 onalizes A. solution. det(A − λI) = (λ − 2) (λ − 8). Therefore A has only two eigen- values1 λ = 2 a2d λ = 8.       −1 −1 1 You may verify thλt= sp 1 , 0 and Eλ = sp 1 1 1 0 1 1 We ﬁnd an orthonormal basis for each of these eigenspaces using the Gram-Schmidt process.  √   √   −1/√2 −1/√6  So  1/ 2 ,−1/ 6 is an orthonormal basλ1 for E  √  0 2/ 6  √   1/ 3   √  and  1/√3  1/ 3 is an orthonormal basis for E 2  √ √ √    −1√ 2 −1/√6 1/ √ 2 0 0 Thus: C = 1/ 2 −1/ 6 1/ 3  and C1AC = 0 2 0 √ √ 0 2/ 6 1/ 3 0 0 8 3 DEFINITION: A linear transformation T : R → R is orthogonal if it n satisﬁes T(v) ▯ T(w) = v ▯ w in R . n n THEOREM: A linear transformation T : R → R is orthogonal ⇐⇒ the standard matrix representation A of T is an orthogonal matrix. proof. of (=⇒) n T is an orthogonal linear transformation of R with standard matrix repre- n sentation A =⇒ for all x,y ∈ R we have: T T T T x ▯ y = T(x) ▯ T(y) = Ax ▯ Ay = (Ax) (Ay) = x (A A)y = x ▯ (A A)y. Thus: T T T n 0 = x ▯ (A Ay) − (x ▯ y) = x ▯ (A Ay − y) = x ▯ (A A − I)y for all x,y ∈ R T If we let x = (A A − I)y we get that: T T T 0 = x ▯ ((A A − I)y) = (A A − I)y ▯ (A A − I)y T T thus (A A − I)y = 0 for all y and so A A = I. proof. (⇐=) If the standard matrix rep of T is the orthogonal matrix A then for all n x,y ∈ R we have: T T T T T T(x) ▯ T(y) = Ax ▯ Ay = (x A )Ay = x (A A)y = x y = x ▯ y 4 EXAMPLE: Determine if T : R −→ R deﬁned by T([x,y,z,w]) =√1 [2x − y,2y + x,2z − w,2w + z] 5 is an orthogonal linear transformation. solution. T is a linear transformation (verify it!) with: √1 T([1,0,0,0]) = [2,1,0,0] 5 T([0,1,0,0]) =√1 [−1,2,0,0] 5 1 T([0,0,1,0]) =√ [0,0,2,1] 5 1 T([0,0,0,1]) =√ [0,0,−1,2] 5  2 −1  √5 √5 0 0  √ √   5 5 0 0  Therefore T has a standard matrix representatio 0 = 0 √ √1 15 2 0 0 √5 √5 T Note that A A = I therefore A is orthogonal and thus T is orthogonal. 5 The Projection Matrix THEOREM: Projection of b on the Subspace W: Let W = sp(a ,a ,.1.a2) k be a k-dimensional subspace of R . Let also, A be the matrix with ith column the column vector a i If b ∈ R , then the projection of b on W is T −1 T bW = A(A A) A b proof. By construction W = column space of A, thus all vectors in W have the form Ax where x ∈ R k We know that b = b W + b W⊥ where b W ∈ W and b W ⊥ ∈ W . Therefore bW = Av for some vector v. In addition, we have that bW⊥ = b − Av, thus b − Av is orthogonal to every vector in W. So for all vectors x ∈ R we have: 0 = Ax▯(b−Av) = (Ax) (b−Av) = x A (b−Av) = x (A b−A Av) = x▯(A b−A Av) T T So: 0 = x ▯ (A b − A Av) for all x ∈ Rk T T T T T −1 T This implies that 0 = (A b−A Av) and A b = A Av or v = (A A) A b T [A A is invertible because A has rank k since it has k linearly independent columns and so A A is a k × k matrix with rank k, thus invertible] Finally, W = Av = A(A A) −1A b as wanted. If we let P = A(A A) −1A then we call P the projection matrix for the subspace W 6 1 Complex Numbers In this lecture we are going to describe a number system C (the “complex numbers”), g√neralizing the real numbers R, and large enough to give a meaning to −1. In fact we have already mentioned them in our discussion of ﬁelds and vector spaces. DEFINITION: The set of complex numbers C is the set of all numbers of the form x + iy where x,y ∈ R equipped with the following operations 1. Addition: If z1= x +1iy ∈1C and z = x 2 iy 2 C th2n we deﬁne def z1+ z 2 x + 1 + i2y + y 1 ∈ C2 2. Multiplication: If z = x + iy ∈ C and z = x + iy ∈ C then we 1 1 1 2 2 2 deﬁne def z1 2= x x1− 2 y +1 2x y + 1 2 ) ∈1C2 NOTE 1: If z = 0+i1 = i then by the above deﬁnition of multiplication, we 2 have: z = zz√= (0+i1)(0+i1) = (0−1)+i(0+0) = −1. Thus we see that 2 i = −1 or −1 = i NOTE 2: It was proven in assignment 1 that this set with these operations is a ﬁeld. NOTE 3: We may deﬁne the scalar multiplication on C by: If r ∈ R and z = x + iy ∈ C the
More Less

Related notes for MATB24H3

OR

Don't have an account?

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.