scholarly journals A class of symmetric and non-symmetric band matrices via binomial coefficients

2021 ◽  
Vol 9 (1) ◽  
pp. 321-330
Author(s):  
Omojola Micheal ◽  
Emrah Kilic
Keyword(s):  
Symmetric Matrix ◽  
Binomial Coefficients ◽  
Band Matrices ◽  
Matrix Classes ◽  
Symmetric Band

Abstract Symmetric matrix classes of bandwidth 2r + 1 was studied in 1972 through binomial coefficients. In this paper, non-symmetric matrix classes with the binomial coefficients are considered where r + s + 1 is the bandwidth, r is the lower bandwidth and s is the upper bandwidth. Main results for inverse, determinants and norm-infinity of inverse are presented. The binomial coefficients are used for the derivation of results.

scholarly journals Evidence of the Poisson/Gaudin–Mehta phase transition for band matrices on global scales

2018 ◽  
Vol 07 (02) ◽  
pp. 1850002
Author(s):  
Sheehan Olver ◽  
Andrew Swan
Keyword(s):  
Phase Transition ◽  
Boundary Conditions ◽  
Rate Of Convergence ◽  
Critical Behavior ◽  
Level Density ◽  
Boundary Effects ◽  
Fourth Moment ◽  
Band Matrices ◽  
Band Matrix ◽  
Symmetric Band

We prove that the Poisson/Gaudin–Mehta phase transition conjectured to occur when the bandwidth of an [Formula: see text] symmetric band matrix grows like [Formula: see text] is naturally observable in the rate of convergence of the level density to the Wigner semi-circle law. Specifically, we show for periodic and non-periodic band matrices the rate of convergence of the fourth moment of the level density is independent of the boundary conditions in the localized regime [Formula: see text] with a rate of [Formula: see text] for both cases, whereas in the delocalized regime [Formula: see text] where boundary effects become important, the rate of convergence for the two ensembles differs significantly, slowing to [Formula: see text] for non-periodic band matrices. Additionally, we examine the case of thick non-periodic band matrices [Formula: see text], showing that the fourth moment is maximally deviated from the Wigner semi-circle law when [Formula: see text], and provide numerical evidence that the eigenvector statistics also exhibit critical behavior at this point.

Inversion of certain symmetric band matrices

1972 ◽  
Vol 12 (1) ◽  
pp. 90-98 ◽  
Author(s):  
Lars Rehnqvist
Keyword(s):  
Band Matrices ◽  
Symmetric Band

Projective congruent symmetric matrices enumeration

2019 ◽  
pp. 204-210
Author(s):  
Olga. A Starikova
Keyword(s):  
Normal Form ◽  
Quadratic Form ◽  
Local Ring ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Symmetric Matrix ◽  
Projective Spaces ◽  
Symmetric Matrices ◽  
Matrix Classes

Projective spaces over local ring R = 2R with principal maximal ideal J; 1+J ⊆ R*2 have been investigated. Quadratic forms and corresponding symmetric matrices A and B are projectively congruent if kA = UBU T for a matrix U ∈ GL(n;R) and for some k ∈ R * : In the case of k = 1 quadratic forms (corresponding symmetric matrices) are called congruent. The problem of enumerating congruent and projective congruent quadratic forms is based on the identification of the (unique) normal form of the corresponding symmetric matrices and is related to the theory of quadratic form schemes. Over the local ring R on conditions R * =R *2 ={1;-1; p;-p} and D(1; 1)=D(1; p)={1; p}; D(1;-1)=D(1;-p)={1;-1; p;-p} (unique) normal form of congruent symmetric matrices over ring R is detected. Quantities of congruent and projective congruent symmetric matrix classes is found when maximal ideal is nilpotent.

scholarly journals Random matrices with exchangeable entries

2020 ◽  
Vol 32 (07) ◽  
pp. 2050022
Author(s):  
Werner Kirsch ◽  
Thomas Kriecherbauer
Keyword(s):  
Asymptotic Behavior ◽  
Random Matrices ◽  
Infinite Sequence ◽  
Random Variables ◽  
Band Matrices ◽  
Exchangeable Random Variables ◽  
Classic Result ◽  
Symmetric Band ◽  
Probability Spaces ◽  
Operator Norms

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding [Formula: see text]-operator norms. The key to our analysis is a generalization of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centered. Some of our results appear to be new even for such Wigner band matrices.

scholarly journals Comparison of eigensolvers for symmetric band matrices

2014 ◽  
Vol 90 ◽  
pp. 55-66 ◽  
Author(s):  
Michael Moldaschl ◽  
Wilfried N. Gansterer
Keyword(s):  
Band Matrices ◽  
Symmetric Band

Some aspects of familiarization with binomial coefficients

2019 ◽  
pp. 49-53
Author(s):  
Abdulkarim Magomedov ◽  
S.A. Lavrenchenko
Keyword(s):  
Elementary Mathematics ◽  
Binomial Coefficients ◽  
Computational Aspects

New laconic proofs of two classical statements of combinatorics are proposed, computational aspects of binomial coefficients are considered, and examples of their application to problems of elementary mathematics are given.

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