A Note on the Spectra of Tridiagonal Matrices

Using orthogonal polynomials, we give a different approach to some recent results on tridi-agonal matrices. 1. Introduction and preliminaries. Most of the results on tridiagonal matrices can be seen in the context of orthogonal polynomials. Here, we will see two examples. The aim of this note is to give simplified proofs of some recent results on tridiagonal matrices , using arguments from the theory of orthogonal polynomials. One of the most important tools in the study of orthogonal polynomials is the spectral theorem for orthogonal polynomials, which states that any orthogonal polynomial sequence (OPS) {P n } n≥0 is characterized by a three-term recurrence relation


Introduction and preliminaries.
Most of the results on tridiagonal matrices can be seen in the context of orthogonal polynomials.Here, we will see two examples.The aim of this note is to give simplified proofs of some recent results on tridiagonal matrices, using arguments from the theory of orthogonal polynomials.
One of the most important tools in the study of orthogonal polynomials is the spectral theorem for orthogonal polynomials, which states that any orthogonal polynomial sequence (OPS) {P n } n≥0 is characterized by a three-term recurrence relation xP n (x) = α n P n+1 (x) + β n P n (x) + γ n P n−1 (x), n = 0, 1, 2,..., ( with initial conditions P −1 (x) = 0 and P 0 (x) = const ≠ 0, where {α n } n≥0 , {β n } n≥0 , and {γ n } n≥0 are sequences of complex numbers such that α n γ n+1 ≠ 0 for all n = 0, 1, 2,....The next proposition is known as the separation theorem for zeros and tells us that the zeros of P n and P n+1 mutually separate each other.
Then there exists a monic orthogonal polynomial sequence (MOPS) {P n } n≥0 such that Notice that the three-term recurrence relation (1.1) can be written in matrix form as . . .
where J n+1 is a tridiagonal Jacobi matrix of order n + 1, defined by (n = 0, 1, 2,...). (1. 3) It follows that if {x nj } n j=1 is the set of zeros of the polynomial P n , then each x nj is an eigenvalue of the corresponding Jacobi matrix J n of order n, and an associated eigenvector is Given a family of orthogonal polynomials {P n } n≥0 defined by (1.1) with α n = 1 for all n and γ n > 0 for all n = 1, 2,... (so that {P n } n≥0 is an MOPS), we may define the associated monic polynomials of order r (r a positive integer) {P (r ) n } n≥0 , r = 0, 1, 2,..., by the shifted recurrence with The anti-associated polynomials for the family {P n } n≥0 , denoted by {P (−r ) n+r } n≥0 , are obtained by pushing down a given Jacobi matrix and by introducing, in the empty upper left corner, new coefficients β −i , i = r ,r − 1,...,1, on the subdiagonal and new coefficients γ −i > 0, i = r − 1,r − 2,...,0, on the lower subdiagonal (see, e.g., [6]).The new Jacobi matrix is then of the form then, clearly, For n > r , the anti-associated polynomials satisfy the three-term recurrence relation The anti-associated polynomial P (−r ) n+r was represented in [6] as a linear combination of the original family P n and the associated polynomials P (1) n−1 in the following way: n−1 (x), n = 0, 1, 2,.... (1.9) 2. Inverse eigenvalue problems.For an n × n matrix A, we denote by σ (A) the spectrum of A and by A the (n−1)th leading principal submatrix of A. In [5], Gray and Wilson stated the following theorem.
Theorem 2.1 [5].Let {µ 1 ,...,µ n } and {ν 1 ,...,ν n−1 } be sets of real numbers satisfying (2.1) Then there exists a unique symmetric tridiagonal n×n matrix A with positive super-and subdiagonals such that This result is a consequence of Theorem 1.2.The positiveness of the two mentioned off-diagonals is a direct consequence of Wendroff's proof.
One says that a set of numbers S is symmetric if S = −S.If {P k } k≥0 is the MOPS given by Wendroff's theorem, then it is possible to prove that β k = 0 for all k in the three-term recurrence relation (1.1).As a consequence, we have the following proposition.

Antipodal tridiagonal pattern.
An n × n (sign) pattern is a matrix, where each entry is +, −, or 0. A pattern S = (s ij ) defines a pattern class of real matrices For n ≥ 2, we consider the antipodal tridiagonal pattern T n defined as In [2], Drew et al. have considered the matrix A n ∈ Q(T n ): They proved that if det(zI n −A n ) = z n , then A n is symmetric about the reverse diagonal.
Theorem 3.1 [2].If there exists This can be generalized in the following way.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: