Controllability,inertia, and stability for tridiagonal matrices

Criteria are given for the controllability of certain pairs of tridiagonal matrices. These criteria may be used, with the Chen-Wimmer theorem, to obtain inertia results. Also, a characterization is given of those nonsingular tridiagonal matrices with certain principal minors nonnegative which are positive stable. This extends a previous characterization of the real D-stable tridiagonal matrices.     

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

A backward stability analysis of diagonal pivoting methods for solving unsymmetric tridiagonal systems without interchanges

This paper concerns the LBM ^T factorization of unsymmetric tridiagonal matrices, where L and M are unit lower triangular matrices and B is block diagonal with 1×1 and 2×2 blocks. In some applications, it is necessary to form this factorization without row or column interchanges while the tridiagonal matrix is formed. Bunch and Kaufman proposed a pivoting strategy without interchanges specifically for symmetric tridiagonal matrices, and more recently, Bunch and Marcia proposed pivoting strategies that are normwise backward stable for linear systems involving such matrices. In this paper, we extend these strategies to the unsymmetric tridiagonal case and demonstrate that the proposed methods both exhibit bounded growth factors and are normwise backward stable. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimates for the inverse elements of tridiagonal matrices

In this work, new upper and lower bounds for the inverse entries of the tridiagonal matrices are presented. The bounds improve the bounds in D. Kershaw [Inequalities on the elements of the inverse of a certain tridiagonal matrix, Math. Comput. 24 (1970) 155–158], P.N. Shivakumar, C.X. Ji [Upper and lower bounds for inverse elements of finite and infinite tridiagonal matrices, Linear Algebr. Appl. 247 (1996) 297–316], R. Nabben [Two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebr. Appl. 287 (1999) 289–305] and R. Peluso, T. Politi [Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear. Algebr. Appl. 330 (2001) 1–14].     

Direct and inverse spectral problems for a class of non-self-adjoint periodic tridiagonal matrices

The spectral properties of a class of tridiagonal matrices are investigated. The reconstruction of matrices of this special class from given spectral data is also studied. Necessary and sufficient conditions for that reconstruction are found. The obtained results extend some results on the direct and inverse spectral problems for periodic Jacobi matrices and for some non-self-adjoint tridiagonal matrices.     

关于三角形Toeplitz系统的复杂性

目前,已有结果表明,作两个n阶上(或下)三角形T矩阵的乘积以及做n阶三角形T矩阵乘n维列向量的算术运算次数,均不超过O(nlog_2n);而求n阶三角形T矩阵的逆,其工作量则不超过O(nlog_2~2n). 本文给出三角形T矩阵求逆与求解三角形Toeplitz线性方程组的快速算法.该算     

Analytical inversion of general periodic tridiagonal matrices

In this paper we present an analytical forms for the inversion of general periodic tridiagonal matrices, and provide some very simple analytical forms which immediately lead to closed formulae for some special cases such as symmetric or perturbed Toeplitz for both periodic and non-periodic tridiagonal matrices. An efficient computational algorithm for finding the inverse of any general periodic tridiagonal matrices from the analytical form is given, it is suited for implementation using Computer Algebra systems such as MAPLE, MATLAB, MACSYMA, and MATHEMATICA. An example is also given to illustrate the algorithm.     

三对角的完全非负矩阵上的Schur-Oppenheim严格不等式

应用完全非负矩阵的 Hadamard中心的性质 ,给出了非奇异三对角完全非负矩阵的Hadamard乘积的行列式的下界估计满足 Schur- Oppenheim严格不等式的充分条件 ,改进了 T.L .Markham的关于三对角的振荡矩阵的相应结果 .     

A note on certain matrices with h(x) – Fibonacci quaternion polynomials

In this paper we consider a g – circulant, right circulant, left circulant and a special kind of a tridiagonal matrices whose entries are h(x) – Fibonacci quaternion polynomials. We present the determinant of these matrices and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x) – Fibonacci quaternion polynomial sequences.     

三对角矩阵的亚正定性

讨论了比三对角矩阵更广泛的一类矩阵的亚正定性,从而给出了三对角矩阵是亚正定矩阵的充分条件.     

A theorem on inverse of tridiagonal matrices

Tridiagonal or Jacobi matrices arise in many diverse branches of mathematics and have been studied extensively. However, there is little written about the inverses of such matrices. In this paper we characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal. The arguments given are elementaryand show that matrices with tridiagonal inverses have an interesting structure.     

Characterization of acyclic d-stable matrices

Necessary and sufficient conditions for D-stability of acyclic matrices are given. Special cases of this characterization are recent results on tridiagonal matrices.     

The Sylvester–Kac matrix space

The Sylvester–Kac matrix is a tridiagonal matrix with integer entries and integer eigenvalues that appears in a variety of applicative problems. We show that it belongs to a four dimensional linear space of tridiagonal matrices that can be simultaneously reduced to triangular form. We name this space after the matrix.     

Inversion of tridiagonal matrices

Summary This paper presents a simple algorithm for inverting nonsymmetric tridiagonal matrices that leads immediately to closed forms when they exist. Ukita's theorem is extended to characterize the class of matrices that have tridiagonal inverses.Journal Paper No. J-10137 of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa. Project 1669, Partial support by National Institutes of Health, Grant GM 13827     

Inverses of banded and k-Hessenberg matrices

Results are obtained on the elements of the inverses of banded and k-Hessenberg matrices. These generalize known results on tridiagonal and Hessenberg matrices.     

On some properties of circulant matrices

A class Σ of matrices is studied which contains, as special subclasses, p-circulant matrices (p ? 1), Toeplitz symmetric matrices and the inverses of some special tridiagonal matrices. We give a necessary and sufficient condition in order that matrices of Σ commute with each other and are closed with respect to matrix product.     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

Fast transforms for tridiagonal linear equations

In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern ²×n ² in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.     

Tridiagonal and upper triangular inverse M-matrices

In this paper we characterize the nonnegative nonsingular tridiagonal matrices belonging to the class of inverse M-matrices. We give a geometric equivalence for a nonnegative nonsingular upper triangular matrix to be in this class. This equivalence is extended to include some reducible matrices.     

分块带状矩阵的逆

1引言如果分块矩阵A=(A_(ij))_(n×n)满足A_(ij)=O(j-i＞p且i-j＞q),其中A_(ij)为m阶矩阵,则称A为(p,q)-分块带状矩阵．分块带状矩阵在一些实际问题中经常出现,例如在量子场论中用途很广的非线性Schr(?)dinger方程的差分离散问题,解热传导问题等,都会遇到分块带状矩阵．常见的分块三对角矩阵,分块五对角矩阵都是特殊的分块带状矩阵．采用通常的方法求解分块带状矩阵的逆矩阵时,需要进行O(n~3)次m阶矩阵的运算．本文首先将分块带状矩阵扩充成可逆的分块上(下)三角矩阵,利用其逆矩阵导出了分块带状矩阵的逆矩阵表达式;进而利用所得到的公式分别推导了分块三对角矩阵及分块五对角矩阵的逆矩阵的快速算法,所需运算量为O(n~2)次m阶矩阵的运算．本文的结果扩充了文[1]等关于分块三对角阵求逆的相关结果．