Inverse Eigenvalue Problems

Transcription

1 Constructing Matrices with Prescribed Eigenvalues N. Jackson Department of Mathematics Math 45 Term Project, Fall 2010

2 Outline Introduction Eigenvalues and Eigenvectors (IEP s) One Simple Algorithm Heuvers Algorithm Proof An Example Benefits and Drawbacks Applications

3 Eigenvalues and Eigenvectors What are Eigenvalues and Eigenvectors? An eigenvalue is any number such that a given square matrix minus that number times the identity matrix has a zero determinant [2]. Ax = λx

4 Eigenvalues and Eigenvectors What are Eigenvalues and Eigenvectors? An eigenvalue is any number such that a given square matrix minus that number times the identity matrix has a zero determinant [2]. Ax = λx

5 (IEP s) (IEP s) A well-studied yet continually developing branch of Linear Algebra concerning construction of matrices from spectral data.[3] Two basic components: solvability and computability.

6 (IEP s) (IEP s) A well-studied yet continually developing branch of Linear Algebra concerning construction of matrices from spectral data.[3] Two basic components: solvability and computability.

7 Heuvers Algorithm Konrad Heuvers Algorithm Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors Let {p 1, p 2,..., p n } be an arbitrary orthonormal basis for R n. These will become the eigenvectors. Let λ 1, λ 2,..., λ n be n arbitrary real numbers (the desired eigenvalues) and τ be any real number such that τ λ j for j = 1, 2,..., n. Define µ j = λ j τ and b j = µ j p j, and let B be the matrix comprised of the column vectors b 1, b 2,...b n. Let S be the matrix S = BB T + τi, a symmetric matrix with the above eigenvectors and eigenvalues.

8 Heuvers Algorithm Konrad Heuvers Algorithm Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors Let {p 1, p 2,..., p n } be an arbitrary orthonormal basis for R n. These will become the eigenvectors. Let λ 1, λ 2,..., λ n be n arbitrary real numbers (the desired eigenvalues) and τ be any real number such that τ λ j for j = 1, 2,..., n. Define µ j = λ j τ and b j = µ j p j, and let B be the matrix comprised of the column vectors b 1, b 2,...b n. Let S be the matrix S = BB T + τi, a symmetric matrix with the above eigenvectors and eigenvalues.

9 Heuvers Algorithm Konrad Heuvers Algorithm Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors Let {p 1, p 2,..., p n } be an arbitrary orthonormal basis for R n. These will become the eigenvectors. Let λ 1, λ 2,..., λ n be n arbitrary real numbers (the desired eigenvalues) and τ be any real number such that τ λ j for j = 1, 2,..., n. Define µ j = λ j τ and b j = µ j p j, and let B be the matrix comprised of the column vectors b 1, b 2,...b n. Let S be the matrix S = BB T + τi, a symmetric matrix with the above eigenvectors and eigenvalues.

10 Heuvers Algorithm Konrad Heuvers Algorithm Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors Let {p 1, p 2,..., p n } be an arbitrary orthonormal basis for R n. These will become the eigenvectors. Let λ 1, λ 2,..., λ n be n arbitrary real numbers (the desired eigenvalues) and τ be any real number such that τ λ j for j = 1, 2,..., n. Define µ j = λ j τ and b j = µ j p j, and let B be the matrix comprised of the column vectors b 1, b 2,...b n. Let S be the matrix S = BB T + τi, a symmetric matrix with the above eigenvectors and eigenvalues.

11 Proof Proof of Heuvers Algorithm The columns of B are (b 1, b 2,..., b n ) = (µ 1 p 1, µ 2 p 2,..., µ n p n ). The rows of B T are of the form µ i p T i. It must be shown that Sp j = λ j p j.

12 Proof Proof of Heuvers Algorithm The columns of B are (b 1, b 2,..., b n ) = (µ 1 p 1, µ 2 p 2,..., µ n p n ). The rows of B T are of the form µ i p T i. It must be shown that Sp j = λ j p j.

13 Proof Proof of Heuvers Algorithm The columns of B are (b 1, b 2,..., b n ) = (µ 1 p 1, µ 2 p 2,..., µ n p n ). The rows of B T are of the form µ i p T i. It must be shown that Sp j = λ j p j.

14 Proof Proof (cont., 2) Sp j = (BB T + τi)p j = BB T p j + τp j µ 1 p T 1 µ 2 p T 2 = [µ 1 p 1, µ 2 p 2,..., µ n p n ]. p j + τp j µ n p T n µ 1 p T 1 p j µ 2 p T 2 = [µ 1 p 1, µ 2 p 2,..., µ n p n ] p j. + τp j µ n p T n p j

15 Proof Proof (cont., 3) Column vector all zeros except p j dotted with itself is one. µ 1 p T 1 p j µ 2 p T 2... = [µ 1 p 1, µ 2 p 2,..., µ n p n ] p j. + τp j µ n p T n p j 0. = [µ 1 p 1, µ 2 p 2,..., µ n p n ] µ j + τp j. 0

16 Proof Proof (cont., 3) Column vector all zeros except p j dotted with itself is one. µ 1 p T 1 p j µ 2 p T 2... = [µ 1 p 1, µ 2 p 2,..., µ n p n ] p j. + τp j µ n p T n p j 0. = [µ 1 p 1, µ 2 p 2,..., µ n p n ] µ j + τp j. 0

17 Proof Proof (cont., 4) = µ 2 j p j + τp j = (µ 2 j + τ)p j = λ j p j We have shown that Sp j = λ j p j, therefore each vector p j and corresponding scalar λ j are an eigenvector and eigenvalue for the matrix S.

18 Proof Proof (cont., 4) = µ 2 j p j + τp j = (µ 2 j + τ)p j = λ j p j We have shown that Sp j = λ j p j, therefore each vector p j and corresponding scalar λ j are an eigenvector and eigenvalue for the matrix S.

19 An Example An Example Arbitrary orthonormal basis for R 2 : {[ ] [ ]} 2/2 2/2 B R 2 =, 2/2 2/2 For simplicity of computing µ 1 and µ 2, we ll choose λ 1 = 2, λ 2 = 5, and τ = 1. µ 1 = λ 1 τ = 2 1 = 1 µ 2 = λ 2 τ = 5 1 = 2

20 An Example An Example Arbitrary orthonormal basis for R 2 : {[ ] [ ]} 2/2 2/2 B R 2 =, 2/2 2/2 For simplicity of computing µ 1 and µ 2, we ll choose λ 1 = 2, λ 2 = 5, and τ = 1. µ 1 = λ 1 τ = 2 1 = 1 µ 2 = λ 2 τ = 5 1 = 2

21 An Example An Example Arbitrary orthonormal basis for R 2 : {[ ] [ ]} 2/2 2/2 B R 2 =, 2/2 2/2 For simplicity of computing µ 1 and µ 2, we ll choose λ 1 = 2, λ 2 = 5, and τ = 1. µ 1 = λ 1 τ = 2 1 = 1 µ 2 = λ 2 τ = 5 1 = 2

22 An Example An Example (cont., 2) Create the matrix B, composed of the columns b 1 and b 2 : B = [b 1, b 2 ] = [µ 1 p 1, µ 2 p 2 ] [ [ ] [ ]] 2/2 2/2 = 1, 2 2/2 2/2 = [ ] 2/2 2 2/2 2

23 An Example An Example (cont., 2) Create the matrix B, composed of the columns b 1 and b 2 : B = [b 1, b 2 ] = [µ 1 p 1, µ 2 p 2 ] [ [ ] [ ]] 2/2 2/2 = 1, 2 2/2 2/2 = [ ] 2/2 2 2/2 2

24 An Example An Example (cont., 3) Now we can create our matrix S: S = BB T + τi [ ] [ ] [ ] 2/2 2 2/2 2/2 1 0 = 2/ [ ] [ ] 5/2 3/2 1 0 = + 3/2 5/2 0 1 [ ] 7/2 3/2 = 3/2 7/2

25 An Example An Example (cont., 4) Check this solution by first finding the eigenvalues of S: det(s λi) = 0 7/2 λ 3/2 3/2 7/2 λ = 0 (7/2 λ) 2 ( 3/2) 2 = 0 49/4 7λ + λ 2 9/4 = 0 λ 2 7λ + 10 = 0 (λ 2)(λ 5) = 0 λ = 2, 5

26 An Example An Example (cont., 5)...and then by finding the corresponding eigenvectors: (S 2I)x = 0 [ ] [ ] [ ] 3/2 3/2 x 0 = 3/2 3/2 y 0 [ ] [ ] x 1 = y 1 (S 5I)x = 0 [ ] [ ] [ ] 3/2 3/2 x 0 = 3/2 3/2 y 0 [ ] [ ] x 1 = y 1

27 An Example An Example (cont., 5)...and then by finding the corresponding eigenvectors: (S 2I)x = 0 [ ] [ ] [ ] 3/2 3/2 x 0 = 3/2 3/2 y 0 [ ] [ ] x 1 = y 1 (S 5I)x = 0 [ ] [ ] [ ] 3/2 3/2 x 0 = 3/2 3/2 y 0 [ ] [ ] x 1 = y 1

28 An Example An Example (cont., 6) Note that the eigenvectors of the S we created with the algorithm are scalar multiples of the eigenvectors we chose beforehand. Since the nullspaces of S 2I and S 5I are both closed under scalar multiplication, the eigenvectors we found confirm the validity of the algorithm.

29 An Example An Example (cont., 6) Note that the eigenvectors of the S we created with the algorithm are scalar multiples of the eigenvectors we chose beforehand. Since the nullspaces of S 2I and S 5I are both closed under scalar multiplication, the eigenvectors we found confirm the validity of the algorithm.

30 An Example An Example (cont., 7) [ ] 1 1 [ ] 2/2 2/2 y [ ] 1 1 [ ] 2/2 x 2/2 Figure: Eigenvectors of matrix S.

31 Benefits and Drawbacks Benefits and Drawbacks of Heuvers Algorithm Simple to understand and compute. Always creates symmetric matrices, must normalize eigenvectors first.

32 Benefits and Drawbacks Benefits and Drawbacks of Heuvers Algorithm Simple to understand and compute. Always creates symmetric matrices, must normalize eigenvectors first.

33 Applications of Found in applications where goal is finding physical parameters of a system based on known behavior or constructing a system with physical parameters resulting in a desired dynamical behavior [3]. Particle physics Molecular spectroscopy Geophysics

34 Applications of Found in applications where goal is finding physical parameters of a system based on known behavior or constructing a system with physical parameters resulting in a desired dynamical behavior [3]. Particle physics Molecular spectroscopy Geophysics

35 Applications of Found in applications where goal is finding physical parameters of a system based on known behavior or constructing a system with physical parameters resulting in a desired dynamical behavior [3]. Particle physics Molecular spectroscopy Geophysics

36 Applications of Found in applications where goal is finding physical parameters of a system based on known behavior or constructing a system with physical parameters resulting in a desired dynamical behavior [3]. Particle physics Molecular spectroscopy Geophysics

37 For Further Reading I G. Strang. Introduction to Linear Algebra, Fourth Edition. Wesley-Cambridge, Trustees of Princeton University WordNet A Lexical Database for English, s=eigenvalue

38 For Further Reading II M. Chu, G. Golub. : Theory and Applications. Department of Mathematics, North Carolina State University, Lectures/Iep/preface.ps K. Heuvers. Symmetric Matrices with Prescribed Eigenvalues and Eigenvectors Mathematics Magazine, Vol. 55, No. 2. (Mar., 1982), pp