Math 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010

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1 Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall / 34

2 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

3 Self-Adjoint Operators Definition An operator T L(V ) is self-adjoint if T = T. Proposition (7.1) Every eigenvalue of a self-adjoint operator is real. Proposition (7.2) If V is a complex inner-product space, and T L(V ), such that then T = 0. Tv, v = 0, for all v V, Not true for real inner-product spaces. Example: T (x, y) = ( y, x). (Tarleton State University) Math 550 Chapter 7 Fall / 34

4 Corollary (7.3) Let V be a complex inner-product space, and let T L(V ). Then T is self-adjoint iff Tv, v R, for every v V. (Tarleton State University) Math 550 Chapter 7 Fall / 34

5 Proposition (7.4) If T is a self-adjoint operator on V such that Tv, v = 0, for all v V, then T = 0. (Tarleton State University) Math 550 Chapter 7 Fall / 34

6 Normal Operators Definition An operator T L(V ) is normal if TT = T T. Note that self-adjoint operators are normal. Example: the operator whose matrix is ( ) wrt. the standard basis in R 3 is normal. (Tarleton State University) Math 550 Chapter 7 Fall / 34

7 Proposition (7.6) An operator T L(V ) is normal iff Tv = T v, for all v V. Corollary (7.7) Suppose T L(V ) is normal. If v V is an eigenvector of T corresponding to λ F, then v is an eigenvector of T corresponding to λ. (Tarleton State University) Math 550 Chapter 7 Fall / 34

8 Corollary (7.8) If T L(V ) is normal, then eigenvectors of T corresponding to distinct eigenvalues are orthogonal. (Tarleton State University) Math 550 Chapter 7 Fall / 34

9 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

10 Goal of Spectral Theorem Reminder An operator T L(V ) has a diagonal matrix wrt. some basis of V iff there is a basis of V consisting of evec. s of T. Goal of Spectral Theorem More ambitious goal: find an orthonormal basis that T has a diagonal matrix wrt. That is, find an orthonormal basis of V consisting of evec. s of T. (Tarleton State University) Math 550 Chapter 7 Fall / 34

11 The Complex Spectral Theorem Complex Spectral Theorem (7.9) Suppose that V is a complex inner-product space, and T L(V ). Then V has an orthonormal basis consisting of evec. s of T iff T is normal. (Tarleton State University) Math 550 Chapter 7 Fall / 34

12 Preliminaries for the Real Spectral Theorem Lemma (7.11) Suppose T L(V ) is self-adjoint. If α, β R, and α 2 < 4β, then T 2 + αt + βi is invertible. Lemma (7.12) If T L(V ) is self-adjoint, then T has an eigenvalue. (Tarleton State University) Math 550 Chapter 7 Fall / 34

13 The Real Spectral Theorem Real Spectral Theorem (7.13) Suppose that V is a real inner-product space, and T L(V ). Then V has an orthonormal basis consisting of evec. s of T iff T is self-adjoint. (Tarleton State University) Math 550 Chapter 7 Fall / 34

14 Decomposing V into Eigenspaces Corollary (7.14) Suppose that T L(V ) is self-adjoint (or that F = C and that T is normal). Let λ 1,..., λ m denote the distinct eval. s of T. Then, V = null(t λ 1 I) null(t λ m I), and each vector in one of the eigenspaces is orthogonal to all vectors in the other eigenspaces. (Tarleton State University) Math 550 Chapter 7 Fall / 34

15 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

16 The 2-dimensional Case Lemma (7.15) Suppose V is a 2-dimensional real inner-product space, and T L(V ). Then TFAE: 1 T is normal but not self-adjoint; 2 the matrix of T wrt. every orthonormal basis of V has the form ( ) a b, b a with b 0; 3 the matrix of T wrt. some orthonormal basis of V has the form ( ) a b, b a with b > 0. (Tarleton State University) Math 550 Chapter 7 Fall / 34

17 Example of Block Matrices A = ( D = D = ( ) A B, where 0 C ) ( 2 2 2, B = ), C = (Tarleton State University) Math 550 Chapter 7 Fall / 34

18 Normal Operators and Invariant Subspaces Proposition Suppose T L(V ) is normal, and U is a subspace of V that is invariant under T. Then, 1 U is invariant under T ; 2 U is invariant under T ; 3 (T U ) = (T ) U ; 4 T U is a normal operator on U; 5 T U is a normal operator on U. (Tarleton State University) Math 550 Chapter 7 Fall / 34

19 Block Diagonal Matrices Definition A square matrix A is block diagonal if there exist square matrices A 1,..., A m, such that A 1 0 A =... 0 A m Multiplying block diagonal matrices: A A m B B m = A 1 B A m B m (Tarleton State University) Math 550 Chapter 7 Fall / 34

20 Normal Operators on Real Inner-Product Spaces Theorem (7.25) Suppose that V is a real inner-product space, and T L(V ). Then, T is normal iff there is an orthonormal basis of V wrt. which T has a block diagonal matrix, where each block is 1 1 or 2 2 of the form ( ) a b, b a with b > 0. (Tarleton State University) Math 550 Chapter 7 Fall / 34

21 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

22 Positive Operators Definition An operator T L(V ) is positive if it is self-adjoint, and for any v V, Tv, v 0. Example Every orthogonal projection is positive. If T is self adjoint, and α, β R such that α 2 < 4β, then T 2 + αt + βi is positive. (Tarleton State University) Math 550 Chapter 7 Fall / 34

23 Square Roots of Operators Definition Suppose T L(V ). If S L(V ) such that S 2 = T, then S is called a square root of T. Theorem (7.27) Let T L(V ). Then TFAE: 1 T is positive; 2 T is self-adjoint and all the evals of T are nonnegative; 3 T has a positive square root; 4 T has a self-adjoint square root; 5 there exists an operator S L(V ) such that T = S S. (Tarleton State University) Math 550 Chapter 7 Fall / 34

24 Proposition (7.28) Every positive operator on V has a unique positive square root. (Tarleton State University) Math 550 Chapter 7 Fall / 34

25 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

26 Isometries Definition An operator S L(V ) is called an isometry if for all v V. Sv = v, Isometries on real inner product spaces are called orthogonal maps. Isometries on complex inner product spaces are called unitary maps. (Tarleton State University) Math 550 Chapter 7 Fall / 34

27 Equivalent Conditions for Isometries Theorem (7.36) Suppose S L(V ). TFAE: 1 S is an isometry; 2 Su, Sv = u, v, for all u, v V ; 3 S S = I; 4 (Se 1,..., Se n ) is orthonormal whenever (e 1,..., e n ) is orthonormal; 5 there exists an orthonormal basis (e 1,..., e n ) of V such that (Se 1,..., Se n ) is orthonormal; 6 S is an isometry. Note that an isometry S is always invertible, and S 1 = S. (Tarleton State University) Math 550 Chapter 7 Fall / 34

28 Example Suppose (e 1,..., e n ) is an o.n. basis for V, and λ 1,..., λ n are scalars with absolute value 1. Suppose Se j = λ j e j, for all j = 1,..., n. Then S is an isometry. Example (Real Case Only) Suppose V is a real inner-product space of dimension 2, and let (e 1, e 2 ) be an o.n. basis for V. Suppose S L(V ), and ( cos θ sin θ Mat(S, (e 1, e 2 )) = sin θ cos θ for some θ R. Then S is an isometry. ), (Tarleton State University) Math 550 Chapter 7 Fall / 34

29 Characterization of Isometries on Complex Inner-Product Spaces Theorem (7.37) Suppose V is a complex inner-product space and S L(V ) Then, S is an isometry iff there exists an o.n. basis of V consisting of eigenvectors of S, corresponding to eigenvalues with absolute value equal to one. (Tarleton State University) Math 550 Chapter 7 Fall / 34

30 Characterization of Isometries on Real Inner-Product Spaces Theorem (7.38) Suppose V is a real inner-product space and S L(V ) Then, S is an isometry iff there exists an o.n. basis of V wrt. which S has a block diagonal matrix, where each block is 1, -1, or a 2 2 block of the form ( ) cos θ sin θ, sin θ cos θ with 0 < θ < π. (Tarleton State University) Math 550 Chapter 7 Fall / 34

31 Outline 1 Self-Adjoint and Normal Operators 2 The Spectral Theorem 3 Normal Operators on Real Inner-Product Spaces 4 Positive Operators 5 Isometries 6 Polar and Singular-Value Decompositions (Tarleton State University) Math 550 Chapter 7 Fall / 34

32 Polar Decomposition Theorem (7.41) If T L(V ), there exists an isometry S L(V ), such that T = S T T. (Tarleton State University) Math 550 Chapter 7 Fall / 34

33 Singular Values Definition Let T L(V ). The singular values of T are the eigenvalues of T T, where each eigenvalue λ is repeated dim null( T T λi) times. Example Define T L(F 4 ) by T (z 1, z 2, z 3, z 4 ) = (0, 3z 1, 2z 2, 3z 4 ). Then, T T (z 1, z 2, z 3, z 4 ) = (3z 1, 2z 2, 0, 3z 4 ). The singular values of T are 3,3,2,0. (Tarleton State University) Math 550 Chapter 7 Fall / 34

34 Singular-Value Decomposition Theorem (7.46) Suppose T L(V ) has singular values s 1,..., s n. Then there exist orthonormal bases (e 1,..., e n ) and (f 1,..., f n ), such that Tv = s 1 v, e 1 f s n v, e n f n, for every v V. In other words, Mat(T, (e 1,..., e n ), (f 1,..., f n )) = s s n (Tarleton State University) Math 550 Chapter 7 Fall / 34

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