On C-Matrices of Arbitrary Powers

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

This paper discusses the factorization of a bent function type complex Hadamard matrix of order N=p/sup n/ with a prime p. The bent function type complex Hadamard matrix is the generalization of the bent function type binary Hadamard matrix. It is shown that the bent function type complex Hadamard matrix is closely related to the complex Hadamard matrix generalized the Sylvester type binary Hadamard matrix, and can be expressed by using 2n-2 matrices of order N with pN non-zero elements at most. The application of the bent function type complex Hadamard matrix to synchronous spread spectrum communication is also discussed.

A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order n all trades contain at least n entries. We call a trade rectangular if it consists of a submatrix that can be multiplied by some scalar c ≠ 1 to obtain another complex Hadamard matrix. We give a characterisation of rectangular trades in complex Hadamard matrices of order n and show that they all contain at least n entries. We conjecture that all trades in complex Hadamard matrices contain at least n entries.KeywordsHadamard matrixTradeRankMathematics Subject Classification (2010):05B2015B34

Path positivity and infinite Coxeter groups

Hadamard matrix is a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. On the other hand, Hadamard-type matrix on finite fields is a similar one, but has multi-valued components on finite fields. To be more specific, we consider n × n matrices H that have their elements on the given finite fields GF(p), and satisfy HHT = nI under modulo p, where I is an identity matrix. Any additions and multiplications should be executed under modulo p. In the authors' previous studies, some properties of Hadamard-type matrices on finite fields have been proven. For example, it has been shown that the order of a Hadamard-type matrix of odd order on GF(p) is limited to a quadratic residue of a given prime p. On the other hand, it is not clear how many and how various Hadamard-type matrices on GF(p) exist in general. In this paper, we count all possible Hadamard-type matrices on finite fields when the order n of a matrix is small. We also categorize Hadamard-type matrices into six different types when n = 3 and p = 11. In addition, we prove that for any prime p, a Hadamard-type matrix over GF(p) in each of the six types always exists if and only if the order ‘3’ is a quadratic residue of a given p.

On the asymptotic existence of partial complex Hadamard matrices and related combinatorial objects

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.

Hadamard matrices play a key role in the study of algebra and quantum information theory, and it is an open problem to characterize 6 6 Hadamard matrices. In this paper, we investigate the problem in terms of the Schmidt rank. The primary achievement of this paper lies in establishing a systematic approach to generate 6 6 Hadamard matrices and H-2 reducible matrices through partial transpose. First, if the Schmidt rank of a Hadamard matrix is at most three, then the partial transpose of the Hadamard matrix is also a Hadamard matrix. Conversely, if the Schmidt rank is four, then the partial transpose is no longer a Hadamard matrix. Second, we discuss the relationship between Schmidt rank and H-2 reducible matrices. We prove Hadamard matrices with Schmidt-rank-one are all H-2 reducible, and prove that some Schmidt-rank-two matrices are H-2 reducible. Finally, we confirm that the partial transpose of an H-2 reducible Schmidt-rank-one or two Hadamard matrix remains H-2 reducible.

This is a presentation of recent work on quantum permutation groups, complex Hadamard matrices, and the connections between them. A long list of problems is included. We include as well some conjectural statements, about matrix models.

Let $G$ be a graph of order $n$ with vertices labeled as $v_1, v_2, \ldots, v_n$. Let $d_{h_i}$ be the hub degree of the vertex $v_i$ for $i=1,2, \ldots, n$. In this paper, we introduce the first hub Zagreb matrix and second hub Zagreb matrix of $G$. The first hub Zagreb matrix of $G$ is the square matrix of order $n$ whose $i j$ th-entry is equal to $d_{h_i}+d_{h_j}$ if $v_i$ is adjacent to $v_j$ and zero otherwise. We further define first hub Zagreb energy and investigate its nature based on the eigenvalues of the first hub Zagreb matrix. In addition, we establish some bounds to first hub Zagreb energy in terms of hub Zagreb index and hyper-hub Zagreb index and calculate the hub Zagreb energy of hydrocarbons. Received: August 10, 2023Accepted: November 9, 2023

Let G be a simple, finite, undirected and connected graph. The eccentricity of a vertex v is the maximum distance from v to all other vertices of G . The eccentricity Laplacian matrix of G with n vertices is a square matrix of order n , whose elements are el ij , where el ij is −1 if the corresponding vertices are adjacent, el ii is the eccentricity of v i for 1 ≤ i ≤ n , and el ij is 0 otherwise. If ǫ1, ǫ2, . . . , ǫn are the eigenvalues of the eccentricity Laplacian matrix, then the eccentricity Laplacian energy of G is ELE(G) = Xn i=1 |ǫi − avec(G)| , where avec(G) is the average eccentricities of all the vertices of G. In this study, some properties of the eccentricity Laplacian energy are obtained and comparison between thge eccentricity Laplacian energy and the total π −electron energy is obtained.

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

The open neighborhood of a vertex u denoted by N(u) in a simple connected graph G is the collection of all vertices other than u and adjacent to u. In this paper, we introduce a square matrix of order n, called the open neighborhood matrix, ONM(G) of a graph G whose (i, j)th entry is |N(vi) ∩ N(vj)|/(di+dj) whenever vi~vj, i≠ j; and zero otherwise, where di and dj, are the degrees of vi and vj respectively. We then establish the relationship between the connectedness of the graph G and the multiplicity of the eigenvalue zero of the matrix ONM(G), if it exists. Furthermore, we found the bounds for the largest open neighbourhood eigenvalue and open neighbourhood energy of graphs.

Throughout the paper, G will denote an additively written, but not always abelian, group of finite order n; and X = (xij) will denote a square matrix of order n with entries from G and whose rows and columns are numbered 0, 1, …, n − 1. We call X a cartesian array (afforded by G) if(1.1) The sequence {−xmi + xki, i = 0,…, n – 1} contains all elements of G whenever k ≠ m.By a theorem of Jungnickel (see Theorem 2.2 in [5]), the transpose of a cartesian array is also a cartesian array. We call G a cartesian group if there is a cartesian array X afforded by G. In this case, we also call (G, X) a cartesian pair.

In this work, we will prove the Cayley-Hamilton theorem using algebraic geometry. We will see a different proof than the one seen in a linear algebra course, in this case we will use the Zariski topology, then we will take advantage of the fact that every square matrix of order n _ n, with entries in a field K, denoted by (aij)n_n can be seen as an element of the affine space of dimension n _ n over the field K and thanks to this, we can resort to algebraic sets and algebraic varieties in order to obtain some results seen in an algebraic geometry and to get a proof of the Cayley-Hamilton theorem.