Orthogonal Matrices with Zero Diagonal. II

C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v ≡ 2 (mod 4), and the skew C-matrices, of order v ≡ 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, λ)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tournaments introduced by Szekeres [24] and by Reid and Brown [21].

Download Full-text

Lower bounds for tau coefficients and operator norms using composite matrix norms

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013034 ◽

1987 ◽

Vol 35 (1) ◽

pp. 49-57

Author(s):

Choon Peng Tan

Keyword(s):

Lower Bounds ◽

General Class ◽

Special Class ◽

Operator Norm ◽

Composite Matrix ◽

Operator Norms ◽

Class Of Matrices ◽

Matrix Norms

Lower bounds for the tau coefficients and operator norms are derived by using composite matrix norms. For a special class of matrices B, our bounds on ‖B‖p (the operator norm of B induced by the ℓp norm) improve upon a general class of Maitre (1967) bounds for p ≥ 2.

Download Full-text

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.

Download Full-text

A Special Class of Matrices (J. F. Foley)

SIAM Review ◽

10.1137/1016094 ◽

1974 ◽

Vol 16 (4) ◽

pp. 548-549

Author(s):

John Z. Hearon

Keyword(s):

Special Class ◽

Class Of Matrices

Download Full-text

Construction of generalized rotations and quasi-orthogonal matrices

Special Matrices ◽

10.1515/spma-2019-0010 ◽

2019 ◽

Vol 7 (1) ◽

pp. 107-113

Author(s):

Luis Verde-Star

Keyword(s):

Image Processing ◽

Hadamard Matrices ◽

Complex Numbers ◽

Data Communications ◽

Matrix Representations ◽

Orthogonal Matrices ◽

Orthogonal Designs ◽

The Matrix ◽

The One

Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

Download Full-text

A Special Class of Matrices

SIAM Review ◽

10.1137/1015098 ◽

1973 ◽

Vol 15 (4) ◽

pp. 787-787

Author(s):

J. F. Foley

Keyword(s):

Special Class ◽

Class Of Matrices

Download Full-text

Complex Hadamard Matrices from Sylvester Inverse Orthogonal Matrices

Open Systems & Information Dynamics ◽

10.1142/s1230161209000281 ◽

2009 ◽

Vol 16 (04) ◽

pp. 387-405 ◽

Cited By ~ 4

Author(s):

Petre Diţă

Keyword(s):

Unit Circle ◽

Hadamard Matrices ◽

Orthogonal Matrices ◽

Novel Method ◽

Free Parameters

A novel method to obtain parametrizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into complex Hadamard matrices, and in this way we find new parametrizations of Hadamard matrices for dimensions n = 8, 10, and 12.

Download Full-text

The recurrent calculation of the inversion of a special class of matrices

Banach Center Publications ◽

10.4064/-3-1-249-253 ◽

1978 ◽

Vol 3 (1) ◽

pp. 249-253

Author(s):

František Mikloško

Keyword(s):

Special Class ◽

Class Of Matrices

Download Full-text

A Lower Bound for the Permanent on a Special Class of Matrices

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-093-2 ◽

1974 ◽

Vol 17 (4) ◽

pp. 529-530

Author(s):

D. J. Hartfiel

Keyword(s):

Lower Bound ◽

Special Class ◽

Class Of Matrices ◽

Image Position

AbstractLet Un(f) denote the class of all n × n (0, 1)-matrices with precisely r-ones, r≥3, in each row and column. Then

Download Full-text

Determinantal Properties of Generalized Circulant Hadamard Matrices

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3836 ◽

2018 ◽

Vol 34 ◽

pp. 639-651

Author(s):

Marilena Mitrouli ◽

Ondrej Turek

Keyword(s):

Numerical Analysis ◽

Hadamard Matrices ◽

Interesting Problem ◽

Challenging Problem ◽

Analytical Formulas ◽

Class Of Matrices ◽

Growth Problem

The derivation of analytical formulas for the determinant and the minors of a given matrix is in general a difficult and challenging problem. The present work is focused on calculating minors of generalized circulant Hadamard matrices. The determinantal properties are studied explicitly, and generic theorems specifying the values of all the minors for this class of matrices are derived. An application of the derived formulae to an interesting problem of numerical analysis, the growth problem, is also presented.

Download Full-text

On Generalized Rotation Matrices

ISRN Applied Mathematics ◽

10.5402/2011/578352 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12

Author(s):

Oskar Maria Baksalary ◽

Götz Trenkler

Keyword(s):

General Class ◽

Rotation Matrices ◽

Class Of Matrices

A general class of matrices, covering, for instance, an important set of proper rotations, is considered. Several characteristics of the class are established, which deal with such notions and properties as determinant, eigenspaces, eigenvalues, idempotency, Moore-Penrose inverse, or orthogonality.

Download Full-text