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  • Characteristic polynomials and eigenvalues

    for a family of tridiagonal real symmetric

    matrices

    Emily Gullerud

    gullerej@uwec.edu

    aBa Mbirika

    mbirika@uwec.edu

    Rita Post

    postrl@uwec.edu

    January 8, 2018

    “The congruence 5 ≡ 3 (mod 4) holds.”, “The polynomial λ2 − 1 has roots ± √

    2.”, and

    “Clearly (−2) · (−1) = 1.” –Post, aBa, and Gullerud’s embarrassments, respectively (2017)

    Abstract

    We explore a certain family {An}∞n=1 of n× n tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of

    this family, we find a closed form solution. The coefficients of these characteristic poly-

    nomials turn out to involve the diagonal entries of Pascal’s triangle in an attractively

    inviting manner. Lastly, we explore a relation between the eigenvalues of various mem-

    bers of the family. More specifically, we give a sufficient condition for when spec(Am)

    is contained in spec(An).

    1

    mailto:mbirika@uwec.edu mailto:postrl@uwec.edu mailto:mbirika@uwec.edu

  • Contents

    1 Introduction 2

    2 Preliminaries and definitions 4

    3 The characteristic polynomials of An 7

    3.1 A recurrence relation for fn(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    3.2 The fn(λ) and their connection to Pascal’s triangle . . . . . . . . . . . . . . 8

    3.3 A closed form fn(λ) for the recurrence relation . . . . . . . . . . . . . . . . . 9

    3.4 Parity properties of the fn(λ) . . . . . . . . . . . . . . . . . . . . . . . . . . 13

    4 The spectrum of An 14

    4.1 Roots of the first seven equations . . . . . . . . . . . . . . . . . . . . . . . . 14

    4.2 A closed form for the roots of fn(λ) . . . . . . . . . . . . . . . . . . . . . . . 15

    4.3 Spectral radius of An and some bounds questions . . . . . . . . . . . . . . . 18

    4.4 Sufficient condition for spec(Am) ⊂ spec(An) . . . . . . . . . . . . . . . . . 19

    5 Open questions 20

    1 Introduction

    Tridiagonal matrices arise in many science and engineering areas for example in parallel

    computing, telecommunication system analysis, and in solving differential equations using

    finite differences [2, 7, 4]. In particular, the eigenvalues of symmetric tridiagonal matrices

    have been studied extensively starting with Golub in 1962 and since then over 40 papers

    have been published on this topic as can be found on MathSciNet [5].

    Tridiagonal symmetric matrices are a subclass of the class of real symmetric matrices.

    A real symmetric matrix is a square matrix A with real-valued entries that are symmetric

    about the diagonal; that is, A equals its transpose AT. Symmetric matrices arise naturally

    in a variety of applications. Real symmetric matrices in particular enjoy the following two

    properties: (1) all of their eigenvalues are real, and (2) the eigenvectors corresponding to dis-

    tinct eigenvectors are orthogonal. In this paper, we investigate the family of real symmetric

    2

  • n× n matrices of the form

    An =

     0 1

    1 . . . . . . . . . . . . 1

    1 0

     with ones on the superdiagonal and subdiagonal and zeroes in every other entry. The two

    main results of our paper are

    • MAIN RESULT 1: a closed form expression for the characteristic polynomial fn(λ) of An and its intimate connection to the diagonals of Pascal’s triangle, and

    • MAIN RESULT 2: an analysis of the spectrum spec(An) of the An matrices and, most importantly, a proof of a closed form for the particular eigenvalues.

    • MAIN RESULT 3: a sufficient condition for when spec(Am) ⊂ spec(An).

    We were initially drawn to this work when noticing the first few characteristic polynomials

    of our An matrices. For example, we give the first seven polynomials below.

    n-value fn(λ)

    1 −1λ 2 1λ2 −1 3 −1λ3 +2λ 4 1λ4 −3λ2 +1 5 −1λ5 +4λ3 −3λ 6 1λ6 −5λ4 +6λ2 −1 7 −1λ7 +6λ5 −10λ3 +4λ

    Color-coding the coefficients to make the connection more transparent, we see that the four

    columns of coefficients correspond directly to the the first four diagonals in Pascal’s trian-

    gle. We used this observation to construct the following formula for the mth characteristic

    polynomial

    fm(λ) =

    bm2 c∑ i=0

    (−1)m+i ( m− i i

    ) λm−2i.

    We prove that this formula satisfies the three-term recurrence formula

    fn(λ) = −λfn−1(λ)− fn−2(λ)

    3

  • with initial conditions f1(λ) = −λ and f2(λ) = λ2 − 1, thereby establishing our first main result. Our second main result was motivated by our observation that the four distinct eigen-

    values of the A4 matrix are the golden ratio, its reciprocal, and their negatives. Furthermore

    via Mathematica, we observed that these four eigenvalues appear in the spectrum of the An

    matrices for the n-values

    4, 9, 14, 19, 24, 29, 34, 39, and 44.

    All of these numbers being congruent to 4 modulo 5 seemed to be no accident. This motivated

    us to prove the following closed form

    λs = 2 cos

    ( sπ

    m+ 1

    ) for s = 1, . . . , n

    for the n distinct eigenvalues of each An matrix, thereby establishing our second main result.

    As an application of this, we use this closed form to give a sufficient condition for when the

    eigenvalues of the matrix Am are contained in the set of eigenvalues of the matrix An, thereby

    establishing our third main result.

    The breakdown of the paper is as follows. In Section 2, we give some preliminaries and

    definitions. In Section 3, we focus on the characteristic polynomials. In Subsection 3.1,

    we derive a three-term recurrence relation for the characteristic polynomials of the family

    of matrices {An}∞n=1. In Subsection 3.2, we unveil a tantalizing connection between the coefficients of the family of characteristic polynomials {fn(λ)}∞n=1 of the matrices {An}∞n=1 and the diagonal columns of Pascal’s triangle. In Subsection 3.3, we prove that our closed

    form satisfies the recurrence relation. We conclude this section by giving a parity property

    of the fn(λ) in Subsection 3.4. In Section 4, we blah...

    Acknowledgments

    The authors thank Pizza Plus in Eau Claire, WI, where we did research at times

    and enjoyed their pizza. We also thank the home of Post’s uncles Rick and Mike whose

    porch hosted our research team as we further investigated this work on Memorial Day

    of 2017. Finally, we thank Phoenix Park where author EmG came up with the awesome

    proof of the spec containments.

    2 Preliminaries and definitions

    A fundamental problem in linear algebra is the determination of the values λ for which the

    set of n homogeneous linear equations in n unknowns Ax = λx has a non-trivial solution.

    4

  • The latter equation may be written in the form (A − λI)x = 0. The general theory of simultaneous linear algebraic equations shows that there is a non-trivial solution if and only

    if the matrix A − λI is singular, that is, det(A − λI) = 0. This leads to the following fundamental definitions used in this paper.

    Definition 2.1 (Eigenvalue and eigenvector). Let A ∈ Cn×n be a square matrix. An eigen- pair of A is a pair (λ,v) ∈ C × (Cn − 0) such that Av = λv. We call λ an eigenvalue and its corresponding nonzero vector v an eigenvector.

    In particular, we often look at the set of eigenvalues of a given matrix A and the largest

    element in absolute value of these eigenvalues defined as follows.

    Definition 2.2 (Spectrum and spectral radius). Let A ∈ Cn×n be a square matrix. The multiset of eigenvalues of A is called the spectrum of A and is denoted spec(A). The spectral

    radius of A is denoted ρ(A) and is given by

    ρ(A) = max{|λ| : λ ∈ spec(A)}.

    The matrices we care about in this paper are a subset of a wider class of symmetric

    matrices called Toeplitz matrices. A Toeplitz matrix is a matrix of the form

    a0 a1 a2 . . . . . . an−1

    a1 a0 a1 . . .

    ...

    a2 a1 . . . . . . . . .

    ... ...

    . . . . . . . . . a1 a2 ...

    . . . a1 a0 a1

    an−1 . . . . . . a2 a1 a0

     .

    In particular, we focus on tridiagonal symmetric matrices. We give the specific definitions

    below.

    Definition 2.3 (Tridiagonal matrix). A tridiagonal symmetric matrix is a Toeplitz matrix

    in which all entries not lying on the diagonal or superdiagonal or subdiagonal are zero.

    We limit our perspective by considering the tridiagonal matrices of the following form.

    Definition 2.4 (The An matrices). Let An be an n × n tridiagonal symmetric matrix in which the diagonal entries are zero, and the superdiagonal and subdiagonal entries are all

    one.

    5

  • Example 2.5. Here are the An matrices for n = 1, . . . , 4.

    A1 = [ 0 ]

    A2 =

    [ 0 1

    1 0

    ] A3 =

     0 1 0

    1 0 1

    0 1 0

     A4 = 

    0 1 0 0

    1 0 1 0

    0 1 0 1

    0 0 1 0

     A typical method of finding the eigenvalues of a square matrix is by