Block tridiagonal matrices arise in applied mathematics, physics, and signal processing. Many applications require knowledge of eigenvalues and eigenvectors of block tridiagonal matrices, which can be prohibitively expensive for large matrix sizes. In this paper, we address the problem of the eigendecomposition of block tridiagonal matrices by studying a connection between their eigenvalues and zeros of appropriate matrix polynomials. We use this connection with matrix polynomials to derive a closed-form expression for the eigenvectors of block tridiagonal matrices, which eliminates the need for their direct calculation and can lead to a faster calculation of eigenvalues. We also demonstrate with an example that our work can lead to fast algorithms for the eigenvector expansion for block tridiagonal matrices.
Chandler-Wilde, Chonchaiya and Lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices ($\pm 1$ on the first sub- and superdiagonal, $0$ everywhere else) is dense in the set of spectra of periodic tridiagonal sign operators on $\ell^2(\mathbb{Z})$. We give a simple proof of this conjecture. As a consequence we get that the set of eigenvalues of tridiagonal sign matrices is dense in the unit disk. In fact, a recent paper further improves this result, showing that this set of eigenvalues is dense in an even larger set.
We analyze the correspondence between finite sequences of finitely supported probability distributions and finite-dimensional, real, symmetric, tridiagonal matrices. In particular, we give an intrinsic description of the topology induced on sequences of distributions by the usual Euclidean structure on matrices. Our results provide an analytical tool with which to study ensembles of tridiagonal matrices, important in certain inverse problems and integrable systems. As an application, we prove that the Euler characteristic of any generic isospectral set of symmetric, tridiagonal matrices is a t…
Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with $-1$'s on the sub-diagonal, $1$'s on the super-diagonal, $-1$ in the $(1,1)$ entry, $1$ in the $(n,n)$ entry and zeros elsewhere. We find the eigen-pairs of the matrices $R_n$.
We find the spectrum and eigenvectors of an arbitrary irreducible complex tridiagonal matrix with two-periodic main diagonal provided that the spectrum and eigenvectors of the matrix with the same sub- and superdiagonals and zero main diagonal is known. Our result substantially generalises some recent results on the Sylvester-Kac matrix and its certain main principal submatrices.
In this paper we express the eigenvalues of real anti-tridiagonal Hankel matrices as the zeros of given rational functions. We still derive eigenvectors for these structured matrices at the expense of prescribed eigenvalues.
Let $n\ge 2$ be an integer. Let $R_n$ denote the $n\times n$ tridiagonal matrix with -1's on the sub-diagonal, 1's on the super-diagonal, -1 in the (1,1) entry, 1 in the (n,n) entry and zeros elsewhere. This paper shows that $R_n$ is closely related to a certain circulant matrix and a certain skew-circulant matrix. More precisely, let $E_n$ denote the exchange matrix which is defined by $E_n(i,j):=δ(i+j,n+1)$. Let $E_+$ (respectively, $E_-$) be the projection defined by $x\mapsto (1/2)(x + E_n x)$ (respectively, $x\mapsto (1/2)(x - E_n x)$). Then $R_n = (π_n - π_n^T) E_+ + (η_n - η_n^T) E_- ,$…
In this paper, the concept of generalized spectral function is introduced for finite-order tridiagonal symmetric matrices (Jacobi matrices) with complex entries. The structure of the generalized spectral function is described in terms of spectral data consisting of the eigenvalues and normalizing numbers of the matrix. The inverse problems from generalized spectral function as well as from spectral data are investigated. In this way, a procedure for construction of complex tridiagonal matrices having real eigenvalues is obtained.
École Normale Supérieure（巴黎高等师范学院）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学学院）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛瓦数学研究所）
We recast Stahl's counterexample from the point of view of the spectral theory of the underlying non-symmetric Jacobi matrices. In particular, it is shown that these matrices are self-adjoint and non-negative in a Krein space and have empty resolvent sets. In fact, the technique of Darboux transformations (aka commutation methods) on spectra which is used in the present paper allows us to treat the class of all G-non-negative tridiagonal matrices. We also establish a correspondence between this class of matrices and the class of signed measures with one sign change. Finally, it is proved that …
Let ${\cal T}_Λ$ be the compact manifold of real symmetric tridiagonal matrices conjugate to a given diagonal matrix $Λ$ with simple spectrum. We introduce {\it bidiagonal coordinates}, charts defined on open dense domains forming an explicit atlas for ${\cal T}_Λ$. In contrast to the standard inverse variables, consisting of eigenvalues and norming constants, every matrix in ${\cal T}_Λ$ now lies in the interior of some chart domain. We provide examples of the convenience of these new coordinates for the study of asymptotics of isospectral dynamics, both for continuous and discrete time.
Random Schroedinger operators with imaginary vector potentials are studied in dimension one. These operators are non-Hermitian and their spectra lie in the complex plane. We consider the eigenvalue problem on finite intervals of length n with periodic boundary conditions and describe the limit eigenvalue distribution when n goes to infinity. We prove that this limit distribution is supported by curves in the complex plane. We also obtain equations for these curves and for the corresponding eigenvalue density in terms of the Lyapunov exponent and the integrated density of states of a "reference…
For any $β>0$, we provide a tridiagonal matrix model and compute the joint eigenvalue density of a random rank one non-Hermitian perturbation of Gaussian and Laguerre $β$-ensembles of random matrices.
