scholarly journals Bidiagonalization of (k, k + 1)-tridiagonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 20-26 ◽  
Author(s):  
S. Takahira ◽  
T. Sogabe ◽  
T.S. Usuda
Keyword(s):  
Tridiagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonalization ◽  
Tridiagonal Matrices ◽  
The Matrix ◽  
Appl Math ◽  
Diagonalization Algorithm

Abstract In this paper,we present the bidiagonalization of n-by-n (k, k+1)-tridiagonal matriceswhen n < 2k. Moreover,we show that the determinant of an n-by-n (k, k+1)-tridiagonal matrix is the product of the diagonal elements and the eigenvalues of the matrix are the diagonal elements. This paper is related to the fast block diagonalization algorithm using the permutation matrix from [T. Sogabe and M. El-Mikkawy, Appl. Math. Comput., 218, (2011), 2740-2743] and [A. Ohashi, T. Sogabe, and T. S. Usuda, Int. J. Pure and App. Math., 106, (2016), 513-523].

scholarly journals The maximal \alpha-index of trees with k pendent vertices and its computation

2020 ◽  
Vol 36 (36) ◽  
pp. 38-46
Author(s):  
Oscar Rojo
Keyword(s):  
Spectral Radius ◽  
Adjacency Matrix ◽  
Diagonal Matrix ◽  
Tridiagonal Matrix ◽  
Tridiagonal Matrices ◽  
Symmetric Tridiagonal Matrix ◽  
The Matrix ◽  
Pendent Vertices ◽  
Equality Holding ◽  
Alpha Index

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

scholarly journals Inversion of Tridiagonal Matrices Using the Dunford-Taylor’s Integral

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 870
Author(s):  
Diego Caratelli ◽  
Paolo Emilio Ricci
Keyword(s):  
Functional Analysis ◽  
Computer Algebra ◽  
Toeplitz Matrices ◽  
Test Cases ◽  
Recursive Procedure ◽  
Tridiagonal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Matrix Eigenvalues ◽  
Complex Valued

We show that using Dunford-Taylor’s integral, a classical tool of functional analysis, it is possible to derive an expression for the inverse of a general non-singular complex-valued tridiagonal matrix. The special cases of Jacobi’s symmetric and Toeplitz (in particular symmetric Toeplitz) matrices are included. The proposed method does not require the knowledge of the matrix eigenvalues and relies only on the relevant invariants which are determined, in a computationally effective way, by means of a dedicated recursive procedure. The considered technique has been validated through several test cases with the aid of the computer algebra program Mathematica©.

scholarly journals A note on the eigenvalue free intervals of some classes of signed threshold graphs

2019 ◽  
Vol 7 (1) ◽  
pp. 218-225
Author(s):  
Milica Anđelić ◽  
Tamara Koledin ◽  
Zoran Stanić
Keyword(s):  
Tridiagonal Matrix ◽  
Constant Sign ◽  
Equitable Partition ◽  
Elementary Matrix ◽  
Matrix Operations ◽  
Threshold Graphs ◽  
The Matrix ◽  
Threshold Graph

Abstract We consider a particular class of signed threshold graphs and their eigenvalues. If Ġ is such a threshold graph and Q(Ġ ) is a quotient matrix that arises from the equitable partition of Ġ , then we use a sequence of elementary matrix operations to prove that the matrix Q(Ġ ) – xI (x ∈ ℝ) is row equivalent to a tridiagonal matrix whose determinant is, under certain conditions, of the constant sign. In this way we determine certain intervals in which Ġ has no eigenvalues.

scholarly journals Alternative proofs of some formulas for two tridiagonal determinants

2018 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Feng Qi ◽  
Ai-Qi Liu
Keyword(s):  
Chebyshev Polynomials ◽  
Tridiagonal Matrix ◽  
Fibonacci Polynomials ◽  
Detailed Proof ◽  
Delannoy Numbers ◽  
Appl Math ◽  
Math Phys

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

On the Matrix With Elements 1/(r+s-1)

1974 ◽  
Vol 17 (2) ◽  
pp. 297-298 ◽  
Author(s):  
W. W. Sawyer
Keyword(s):  
Tridiagonal Matrix ◽  
The Matrix ◽  
Eigenvectors And Eigenvalues

A standard example of a matrix for which the computation of eigenvectors and eigenvalues is very awkward is the matrix A with ars=1/(r+s—1), 1≤r≤n, 1 ≤s≤n. It is therefore of interest that A commutes with a tridiagonal matrix.

Rank, trace, eigenvalues and norms of a structured matrix

2019 ◽  
Vol 12 (06) ◽  
pp. 2040015
Author(s):  
Ahmet İpek
Keyword(s):  
Real Sequence ◽  
Simple Property ◽  
Structured Matrix ◽  
The Matrix ◽  
Matrix Formula ◽  
Appl Math ◽  
Real Matrices

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

A Nonunitary Joint Block Diagonalization Algorithm for Blind Separation of Convolutive Mixtures of Sources

2007 ◽  
Vol 14 (11) ◽  
pp. 860-863 ◽  
Author(s):  
Hicham Ghennioui ◽  
Fadaili El Mostafa ◽  
NadÈge Thirion-Moreau ◽  
Abdellah Adib ◽  
Eric Moreau
Keyword(s):  
Block Diagonalization ◽  
Blind Separation ◽  
Convolutive Mixtures ◽  
Diagonalization Algorithm

scholarly journals Two-Stage Cubature Kalman Filtering Based on T-Transform and Its Application

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Lu Zhang ◽  
Qiugen Xiao ◽  
Hailun Wang ◽  
Yinyin Hou
Keyword(s):  
Kalman Filtering ◽  
Equivalent Transformation ◽  
Block Diagonalization ◽  
Tracking Accuracy ◽  
Two Stage ◽  
Filtering Algorithm ◽  
Actual Application ◽  
Wheeled Robot ◽  
The Matrix ◽  
Kalman Filtering Algorithm

According to the actual application system model which has bias, this paper analyzes the shortage of the conventional augmented algorithm, the two-stage cubature Kalman filtering algorithm, which is presented on the basis of a two-stage nonlinear transformation. The core ideas of the algorithm are to obtain the block diagonalization of the covariance matrix using the matrix transformation and avoid calculating the covariance of the state and bias to reduce the amount of calculation and ensure a smooth filtering process. Then, the equivalence of the two-stage cubature Kalman filtering algorithm and the cubature Kalman filtering algorithm is proved by updating equivalent transformation. Through the experiment of trajectory tracking of a wheeled robot, it is verified that the two-stage cubature Kalman filtering algorithm can obtain good tracking accuracy and stability with the presence of unknown random bias. Simultaneously, the equivalence of the two-stage cubature Kalman filtering algorithm and cubature Kalman filtering algorithm is verified again using the contrast experiment.

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