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Eigenvalues of a Special Tridiagonal Matrix

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Eigenvalues of a Special Tridiagonal Matrix Eigenvalues of a Special Tridiagonal Matrix Alexander De Serre Rothney∗ October 10, 2013 Abstract In this paper we consider a special tridiagonal test matrix. We prove that its eigenvalues are the even integers 2;:::; 2n and show its relationship with the famous Kac-Sylvester tridiagonal matrix. 1 Introduction We begin with a quick overview of the theory of symmetric tridiagonal matrices, that is, we detail a few basic facts about tridiagonal matrices. In particular, we describe the symmetrization process of a tridiagonal matrix as well as the orthogonal polynomials that arise from the characteristic polynomials of said matrices. Definition 1.1. A tridiagonal matrix, Tn, is of the form: 2a1 b1 0 ::: 0 3 6 :: :: : 7 6c1 a2 : : : 7 6 7 6 :: :: :: 7 Tn = 6 0 : : : 0 7 ; (1.1) 6 7 6 : :: :: 7 4 : : : an−1 bn−15 0 ::: 0 cn−1 an where entries below the subdiagonal and above the superdiagonal are zero. If bi 6= 0 for i = 1; : : : ; n − 1 and ci 6= 0 for i = 1; : : : ; n − 1, Tn is called a Jacobi matrix. In this paper we will use a more compact notation and only describe the subdiagonal, diagonal, and superdiagonal (where appropriate). For example, Tn can be rewritten as: 0 1 b1 : : : bn−1 Tn = @ a1 a2 : : : an−1 an A : (1.2) c1 : : : cn−1 ∗Bishop's University, Sherbrooke, Quebec, Canada 1 Note that the study of symmetric tridiagonal matrices is sufficient for our purpose as any Jacobi matrix with bici > 0 8i can be symmetrized through a similarity transformation: 0 p p 1 b1c1 ::: bn−1cn−1 −1 An = Dn TnDn = @ a1 a2 : : : an−1 an A ; (1.3) p p b1c1 ::: bn−1cn−1 r cici+1 ··· cn−1 where Dn = diag(γ1; : : : ; γn) and γi = . bibi+1 ··· bn−1 We refer the reader to [1] for a proof and more detailed exposition. The added symmetry allows for an easier analysis of the spectrum of An. In particular, a cofactor expansion along the last row of Pn = An − λIn yields the recurrence relations: P0(λ) = 1 (1.4) P1(λ) = a1 − λ (1.5) Pi(λ) = (ai − λ)Pi−1(λ) − bi−1ci−1Pi−2(λ): (1.6) Here fPig is an orthogonal family of polynomials with respect to the inner product: Z +1 hPn;Pmi := Pn(x)Pm(x)w(x)dx; (1.7) −∞ where w(x) is the measure or weight function w(x) = e−x2 . Orthogonality yields the following useful properties (see [9]) : The zeros of Pi are real, 1 ≤ i ≤ n; (1.8) The zeros of Pi and Pi+1 interlace, 1 ≤ i ≤ n − 1: (1.9) In other words, the eigenvalues of An are real and the eigenvalues of Ai−1 interlace those of Ai for 1 ≤ i ≤ n. An interesting problem in matrix theory is that of the inverse eigenvalue problem (IEP). Before formally stating the problem for tridiagonal matrices, let us introduce some notation. Definition 1.2. Given Tn an n×n tridiagonal matrix, the (n−1)×(n−1) principal submatrix, T^n, is the matrix formed by removing the last row and column of Tn. n n−1 IEP for Tridiagonal Matrices. Given the ordered lists Λ = (λi)i=1 and Θ = (θi)i=1 such that Θ interlaces Λ, i.e., λi ≤ θi ≤ λi+1 for i = 1; : : : ; n, find the (n × n) symmetric tridiagonal matrix Tn such that Λ and Θ are the spectra of Tn and T^n, respectively. Note that the existence and uniqueness (up to signs) of Tn from spectral data is only guaranteed when Λ and Θ strictly interlace; we refer the reader to [7] for more details. Also, the IEP for tridiagonal matrices is fully solved in the sense that given the lists Λ and Θ, one can reconstruct Tn algorithmically (see [8], [5, page 473]). In this paper, we are interested in the tridiagonal test matrix Wn that has spectrum Λ = f2; 4;:::; 2ng and W^n has spectrum Θ = f3; 5;:::; 2n − 1g. By test matrix we mean a matrix with known eigenvalues and given structure. Such matrices make it possible to test the stability of numerical eigenvalue algorithms. The motivation behind Wn is provided in section 2. A famous tridiagonal matrix is the Kac-Sylvester matrix proposed by Clement [2] as a test matrix. 2 Definition 1.3. The (n + 1) × (n + 1) Kac-Sylvester matrix, Kn, is: 0 n n − 1 ::: 2 1 1 Kn = @ 0 0 :::::: 0 0 A : (1.10) 1 2 : : : n − 1 n n It has the particularly nice eigenvalues : σ(Kn) = f2k − ngk=0. There are several proofs that Kn has the above spectrum (see [3], [6], [10] ). The relevance of this matrix will become apparent when we prove our main result. 2 Motivation The problem of finding the tridiagonal matrix Tn with the spectrum of Tn being Λ = f2; 4;:::; 2ng and the spectrum of T^n being Θ = f3; 5;:::; 2n − 1g was posed by one of my research supervisors, Dr. N. B. Willms. It arises from the study of spring-mass systems in free motion, where the eigenvalues correspond to natural frequencies of the systems. It turns out that many spring-mass systems beget tridiagonal matrices (see [4]), where the entries of the corresponding tridiagonal matrix are functions of the spring constants and masses of the systems. n More specifically, given n masses fmigi=1 in suspension from a ceiling with the respective spring n−1 constants fkigi=0 , where the hanging end of the system is free (called a fixed-free system), we wish to model this system in terms of matrices. The solutions of jλM − EKE−1j = 0 are precisely the natural frequencies of the system (see [4, page 45]), where M, K, and E are given by: M = diag(m1; m2; : : : ; mn−1; mn): (2.1) 0 1 −k1 ::: −kn−1 K = @ k0 + k1 k1 + k2 : : : kn−2 + kn−1 kn−1 A ; (2.2) −k1 ::: −kn−1 21 −1 0 :::::: 0 3 21 1 :::::: 1 13 6 :: :: 7 6 :: :: 7 60 1 −1 : : 0 7 60 1 1 : : 17 6 7 6 7 6 :: :: :: : 7 6 :: :: :: :7 60 0 : : : : 7 −1 60 0 : : : :7 E = 6 7 and E = 6 7 ; (2.3) 6: :: :: :: 7 6: :: :: :: :7 6: : : : −1 0 7 6: : : : 1 :7 6 7 6 7 6: : : 7 6: 7 4: :: :: 0 1 −15 4:::::::: 0 1 15 0 :::::: 0 0 1 0 :::::: 0 0 1 Note that E is upper-bidiagonal and that E−1 is upper-triangular with all ones. As such our problem is to verify the form of the symmetric tridiagonal matrix B = L−1EKE−1L−T (with p p p L = diag( m1; m2;:::; mn)) with eigenvalues Λ and such that σ(B^) = Θ. As mentioned earlier, there exists an algorithm for the IEP for tridiagonal matrices. Our problems differs from the former as we wish to find Tn with entries as explicit functions of n. 3 3 Main Result We begin with a definition of the matrix of interest which we shall show to be the solution of the IEP. Definition 3.1. Let Wn(k) be the n × n symmetric tridiagonal matrix with the following entries: 8 >ai = k; i = 1; : : : ; n <> q W (k) = b = i(2n−1−i) ; i = 1; : : : ; n − 2 n i p 4 > n(n−1) :>b = p ; n−1 2 as per definition (1.1). For example, 2 q 5 3 7 2 0 0 0 0 6q q 7 6 5 7 9 0 0 0 7 6 2 2 7 6 q q 7 6 9 12 7 6 0 2 7 2 0 0 7 W6(7) = 6 q q 7 : 6 12 14 7 6 0 0 2 7 2 0 7 6 q q 7 6 14 30 7 6 0 0 0 2 7 2 7 4 5 q 30 0 0 0 0 2 7 We are now ready to introduce and prove our main result: Theorem 3.2. The spectra of Wn(n + 1) and W^ n(n + 1) are Λ = f2; 4;:::; 2ng and Θ = f3; 5;:::; 2n − 1g, respectively. Proof. Part A - Eigenvalues of Wn(n + 1) By the Schur decomposition theorem, there exists unitary Q 2 Mn×n(C) such that −1 Q Wn(n + 1)Q = U is upper-triangular with the eigenvalues of Wn(n + 1) being on the diagonal of U. Hence, subtracting (n + 1)In from Wn(n + 1), where In is the n × n identity matrix, shifts the eigenvalues by −(n + 1). Note that Wn(n + 1) − (n + 1)In = Wn(0), the (n × n) symmetric tridiagonal matrix with zero diagonal. Now, choose H = diag(γ1; : : : ; γn) such that, 0 1 2 : : : n − 2 2(n − 1) 1 ~ −1 Wn(0) := 2HWn(0)H = @0 0 :::::: 0 0A : (3.1) 2n − 2 2n − 3 : : : n + 1 n th th That is, given that the i ; (i + 1) (as per definition (1.1)), i = 1; : : : ; n − 2, entry of Wn(0) is bi, the corresponding entry of HW (0)H−1 on the superdiagonal is γi b . We find γ ,γ such that n γi+1 i i i+1 γi b = i for i = 1; : : : ; n − 2. Note that b introduces an extra factor of 2 so that γn−1 b γi+1 i 2 n−1 γn n−1 is twice as large as the pattern suggests. In other words γn−1 b = n − 1. On the other hand, γn n−1 for the subdiagonal, since γi b = i , we have that γi+1 = 2 b and thus γi+1 b = 2 b2.
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