In this paper we study a one-dimensional quantum random walk with the Hadamard transformation which is often called the Hadamard walk. We construct the Hadamard walk using a transition matrix on probability amplitude and give some results on symmetricity of probability distributions for the Hadamard walk.

In 1867, Syvlester noted that the Kronecker product of two Hadamard matrices is an Hadamard matrix. This gave a way to obtain an Hadamard matrix of exponent four from two of exponent two. Early last decade, Agayan and Sarukhanyan found a way to combine two Hadamard matrices of exponent two to obtain one of exponent three, and just last year Craigen, Seberry and Zhang discovered how to combine f...

There is a well-known correspondence between symmetric 2-(4ju— 1,2/z — 1,/z — 1) designs and Hadamard matrices of order 4^. It is not so well known that there is a natural correspondence between Hadamard matrices of order 2\x and affine l-(4ju, 2^, 2(i) designs whose duals are also affine. Such a design is denoted by H^fi) in this paper. Under this natural correspondence, two H^fi) designs are ...

Hadamard matrices with a subjacent algebraic structure have been deeply studied as well as the links with other topics in algebraic combinatorics [1]. An important and pioneering paper about this subject is [5], where it is introduced the concept of Hadamard group. In addition, we find beautiful equivalences between Hadamard groups, 2-cocyclic matrices and relative difference sets [4], [7]. Fro...

In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix is cocyclic is describe...

It is well-known that a Sylvester Hadamard matrix can be described by means of a cocycle. In this paper it is shown that the additional internal structure in a Sylvester Hadamard matrix, provided by the cocycle, is su cient to guarantee some kind of reduction in the computational complexity of calculating its permanent using Ryser's formula. A Hadamard matrix H of order n is an n× n matrix with...

in this paper we establish several polynomials similar to bernstein's polynomials and several refinements of  hermite-hadamard inequality for convex functions.

New constructions for amicable orthogonal designs are given. These new designs then give new amicable Hadamard matrices and new skew-Hadamard matrices. In particular we show that if p is the order of normalized amicable Hadamard matrices there are normalized amicable Hadamard matrices of order (p 1)u + 1, u > 0 an odd integer. Tables are given for the existence of amicable and skew-Hadamard mat...

We prove that if there exist Hadamard matrices of order 4m, 4n, 4p, 4q then there exists an Hadamard matrix of order 16mnpq. This improves and extends the known result of Agayan that there exists an Hadamard matrix of order 8mn if there exist Hadamard matrices of order 4m and 4n.

‎This paper presents two main results that the singular values of the Hadamard product of normal matrices $A_i$ are weakly log-majorized by the singular values of the Hadamard product of $|A_{i}|$ and the singular values of the sum of normal matrices $A_i$ are weakly log-majorized by the singular values of the sum of $|A_{i}|$‎. ‎Some applications to these inequalities are also given‎. ‎In addi...