scholarly journals Block Representation and Spectral Properties of Constant Sum Matrices

2018 ◽  
Vol 34 ◽  
pp. 170-190 ◽  
Author(s):  
Sally Hill ◽  
Matthew Lettington ◽  
Karl Michael Schmidt
Keyword(s):  
Spectral Properties ◽  
Structured Matrices ◽  
Eigenvalues And Eigenvectors ◽  
Equivalent Representation ◽  
Symmetry Properties ◽  
Explicit Formulae ◽  
Block Structured

An equivalent representation of constant sum matrices in terms of block-structured matrices is given in this paper. This provides an easy way of constructing all constant sum matrices, including those with further symmetry properties. The block representation gives a convenient description of the dihedral equivalence of such matrices. It is also shown how it can be used to study their spectral properties, giving explicit formulae for eigenvalues and eigenvectors in special situations, as well as for quasi-inverses when these exist.

scholarly journals A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices

2016 ◽  
Vol 2016 ◽  
pp. 1-3 ◽  
Author(s):  
Hal Caswell ◽  
Silke F. van Daalen
Keyword(s):  
Simple Formula ◽  
Column Vector ◽  
Structured Matrices ◽  
Structured Matrix ◽  
Block Structured

The vec operator transforms a matrix to a column vector by stacking each column on top of the next. It is useful to write the vec of a block-structured matrix in terms of the vec operator applied to each of its component blocks. We derive a simple formula for doing so, which applies regardless of whether the blocks are of the same or of different sizes.

scholarly journals Well-solvable cases of the QAP with block-structured matrices

2015 ◽  
Vol 186 ◽  
pp. 56-65 ◽  
Author(s):  
Eranda Çela ◽  
Vladimir G. Deineko ◽  
Gerhard J. Woeginger
Keyword(s):  
Structured Matrices ◽  
Block Structured

scholarly journals ON THE SPECTRAL PROPERTIES OF MATRICES ASSOCIATED WITH TREND FILTERS

2010 ◽  
Vol 26 (4) ◽  
pp. 1247-1261 ◽  
Author(s):  
Alessandra Luati ◽  
Tommaso Proietti
Keyword(s):  
Spectral Properties ◽  
Eigenvalues And Eigenvectors ◽  
Analytical Results ◽  
Eigen Decomposition

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the circulant and the generalized tau. These results are used to characterize the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.

Backward Error Analysis for an Eigenproblem Involving Two Classes of Matrices

2014 ◽  
Vol 4 (4) ◽  
pp. 329-344
Author(s):  
Lei Zhu ◽  
Weiwei Xu
Keyword(s):  
Error Analysis ◽  
Structured Matrices ◽  
Backward Error Analysis ◽  
Backward Error ◽  
Numerical Examples ◽  
Classes Of Matrices ◽  
Explicit Formulae ◽  
Backward Errors

AbstractWe consider backward errors for an eigenproblem of a class of symmetric generalised centrosymmetric matrices and skew-symmetric generalised skew-centrosymmetric matrices, which are extensions of symmetric centrosymmetric and skew-symmetric skew-centrosymmetric matrices. Explicit formulae are presented for the computable backward errors for approximate eigenpairs of these two kinds of structured matrices. Numerical examples illustrate our results.

Spectral Properties of Circulant Quantum Markov Semigroups

2014 ◽  
Vol 21 (04) ◽  
pp. 1450007 ◽  
Author(s):  
Jorge R. Bolaños-Servin ◽  
Raffaella Carbone
Keyword(s):  
Explicit Expression ◽  
Spectral Properties ◽  
Infinitesimal Generator ◽  
Spectral Gap ◽  
Markov Semigroups ◽  
Eigenvalues And Eigenvectors ◽  
Commutative Algebras

We study the spectral properties of the generators of circulant quantum Markov semigroups. We can find an explicit expression for eigenvalues and eigenvectors of the infinitesimal generator and, in particular, we prove that the spectral gap is strictly positive. By proper techniques, we can reduce the problem on non-commutative algebras to the analogous one for a classical process with a circulant generator.

scholarly journals SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250003 ◽  
Author(s):  
RIDDHIPRATIM BASU ◽  
ARUP BOSE ◽  
SHIRSHENDU GANGULY ◽  
RAJAT SUBHRA HAZRA
Keyword(s):  
Spectral Distribution ◽  
Spectral Properties ◽  
Triangular Matrices ◽  
Wigner Matrix ◽  
Limiting Spectral Distribution ◽  
Convergence Of Sequences ◽  
Explicit Formulae ◽  
Patterned Matrices

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

scholarly journals Eigenproblem for Jacobi matrices: hypergeometric series solution

2007 ◽  
Vol 366 (1867) ◽  
pp. 1089-1114 ◽  
Author(s):  
V.B Kuznetsov ◽  
E.K Sklyanin
Keyword(s):  
Inversion Formula ◽  
Series Solution ◽  
Hypergeometric Series ◽  
Jacobi Matrix ◽  
Algebraic Equations ◽  
Eigenvalues And Eigenvectors ◽  
Principal Diagonal ◽  
Power Series Expansions ◽  
Explicit Formulae ◽  
Jacobi Determinant

We study the perturbative power series expansions of the eigenvalues and eigenvectors of a general tridiagonal (Jacobi) matrix of dimension d . The (small) expansion parameters are the entries of the two diagonals of length d −1 sandwiching the principal diagonal that gives the unperturbed spectrum. The solution is found explicitly in terms of multivariable (Horn-type) hypergeometric series in 3 d −5 variables in the generic case. To derive the result, we first rewrite the spectral problem for the Jacobi matrix as an equivalent system of algebraic equations, which are then solved by the application of the multivariable Lagrange inversion formula. The corresponding Jacobi determinant is calculated explicitly. Explicit formulae are also found for any monomial composed of eigenvector's components.

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