G-matrices, J-orthogonal matrices, and their sign patterns

Abstract This paper builds upon the results in the article “G-matrices, J-orthogonal matrices, and their sign patterns", Czechoslovak Math. J. 66 (2016), 653-670, by Hall and Rozloznik. A number of further general results on the sign patterns of the J-orthogonal matrices are proved. Properties of block diagonal matrices and their sign patterns are examined. It is shown that all 4 × 4 full sign patterns allow J-orthogonality. Important tools in this analysis are Theorem 2.2 on the exchange operator and Theorem 3.2 on the characterization of J-orthogonal matrices in the paper “J-orthogonal matrices: properties and generation", SIAM Review 45 (3) (2003), 504-519, by Higham. As a result, it follows that for n ≤4 all n×n full sign patterns allow a J-orthogonal matrix as well as a G-matrix. In addition, the 3 × 3 sign patterns of the J-orthogonal matrices which have zero entries are characterized.

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Sign patterns of orthogonal matrices and the strong inner product property

Linear Algebra and its Applications ◽

10.1016/j.laa.2020.01.033 ◽

2020 ◽

Vol 592 ◽

pp. 228-259

Author(s):

Bryan A. Curtis ◽

Bryan L. Shader

Keyword(s):

Inner Product ◽

Orthogonal Matrices ◽

Product Property ◽

Sign Patterns

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Sign patterns occurring in orthogonal matrices

Linear and Multilinear Algebra ◽

10.1080/03081089808818565 ◽

1998 ◽

Vol 44 (4) ◽

pp. 287-299 ◽

Cited By ~ 4

Author(s):

Charles R. Johnson ◽

Charles Waters ◽

S. Pierce

Keyword(s):

Orthogonal Matrices ◽

Sign Patterns

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Nested code sequences of Barker — Mersenne — Raghavarao

Information and Control Systems ◽

10.31799/1684-8853-2019-3-71-81 ◽

2019 ◽

pp. 71-81

Author(s):

M. B. Sergeev ◽

V. A. Nenashev ◽

A. M. Sergeev

Keyword(s):

Data Transfer ◽

Radio Channel ◽

Main Peak ◽

Synthesis Methods ◽

Orthogonal Matrices ◽

Useful Signal ◽

New Codes ◽

Increase In Accuracy ◽

Application Fields ◽

Barker Codes

Introduction: The problem of noise-free encoding for an open radio channel is of great importance for data transfer. The results presented in this paper are aimed at stimulating scientific interest in new codes and bases derived from quasi-orthogonal matrices, as a basis for the revision of signal processing algorithms.Purpose: Search for new code sequences as combinations of codes formed from the rows of Mersenne and Raghavarao quasi-orthogonal matrices, as well as complex and more efficient Barker — Mersenne — Raghavarao codes.Results: We studied nested code sequences derived from the rows of quasi-orthogonal cyclic matrices of Mersenne, Raghavarao and Hadamard, providing estimates for the characteristics of the autocorrelation function of nested Barker, Mersenne and Raghavarao codes, and their combinations: in particular, the ratio between the main peak and the maximum positive and negative “side lobes”. We have synthesized new codes, including nested ones, formed on the basis of quasi-orthogonal matrices with better characteristics than the known Barker codes and their nested constructions. The results are significant, as this research influences the establishment and development of methods for isolation, detection and processing of useful information. The results of the work have a long aftermath because new original code synthesis methods need to be studied, modified, generalized and expanded for new application fields.Practical relevance: The practical application of the obtained results guarantees an increase in accuracy of location systems, and detection of a useful signal in noisy background. In particular, these results can be used in radar systems with high distance resolution, when detecting physical objects, including hidden ones.

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Circles on lattices and Hadamard matrices

Information and Control Systems ◽

10.31799/1684-8853-2019-3-2-9 ◽

2019 ◽

pp. 2-9 ◽

Cited By ~ 1

Author(s):

N. A. Balonin ◽

M. B. Sergeev ◽

J. Seberry ◽

O. I. Sinitsyna

Keyword(s):

Hadamard Matrices ◽

Lattice Points ◽

Practical Significance ◽

Upper And Lower Bounds ◽

Problem Size ◽

Orthogonal Matrices ◽

Video Information ◽

Scientific Schools ◽

Gauss Theorem ◽

Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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Exact Moment Simulation Using Random Orthogonal Matrices

SSRN Electronic Journal ◽

10.2139/ssrn.1466709 ◽

2009 ◽

Author(s):

Carol Alexander ◽

Walter Ledermann ◽

Dan Ledermann

Keyword(s):

Orthogonal Matrices ◽

Exact Moment

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Sarnak’s conjecture for sequences of almost quadratic word growth

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.94 ◽

2020 ◽

pp. 1-56

Author(s):

REDMOND MCNAMARA

Keyword(s):

Box Dimension ◽

Multiplicative Functions ◽

Logarithmic Density ◽

Liouville Function ◽

Sign Patterns

Abstract We prove the logarithmic Sarnak conjecture for sequences of subquadratic word growth. In particular, we show that the Liouville function has at least quadratically many sign patterns. We deduce the main theorem from a variant which bounds the correlations between multiplicative functions and sequences with subquadratically many words which occur with positive logarithmic density. This allows us to actually prove that our multiplicative functions do not locally correlate with sequences of subquadratic word growth. We also prove a conditional result which shows that if the ( $\kappa -1$ )-Fourier uniformity conjecture holds then the Liouville function does not correlate with sequences with $O(n^{t-\varepsilon })$ many words of length n where $t = \kappa (\kappa +1)/2$ . We prove a variant of the $1$ -Fourier uniformity conjecture where the frequencies are restricted to any set of box dimension less than $1$ .

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