Parametrizing Complex Hadamard Matrices

The purpose of this paper is to introduce new parametric families of complex Hadamard matrices in two different ways. First, we prove that every real Hadamard matrix of order N>=4 admits an affine orbit. This settles a recent open problem of Tadej and Zyczkowski, who asked whether a real Hadamard matrix can be isolated among complex ones. In particular, we apply our construction to the only (up to equivalence) real Hadamard matrix of order 12 and show that the arising affine family is different from all previously known examples. Second, we recall a well-known construction related to real conference matrices, and show how to introduce an affine parameter in the arising complex Hadamard matrices. This leads to new parametric families of orders 10 and 14. An interesting feature of both of our constructions is that the arising families cannot be obtained via Dita's general method. Our results extend the recent catalogue of complex Hadamard matrices, and may lead to direct applications in quantum-information theory.Comment: 16 pages; Final version. Submitted to: European Journal of Combinatoric

Some orthogonal matrices constructed by strong Kronecker multiplication

Strong Kronecker multiplication of two matrices is useful for constructing new orthogonal matrices from those known. These results are particularly important as they allow small matrices to be combined to form larger matrices, but of smaller order than the straight-forward Kronecker product would permit

Some Orthogonal Matrices Constructed by Strong Kronecker Multiplication

Strong Kronecker multiplication of two matrices is useful for constructing new orthogonal matrices from known those. In this paper we give strong Kronecker multiplication a general form and a short proof. To show its applications, we prove that if there exists a complex Hadamard matrix of order 2c then there exists (i) a W (4nc; 2kc), if there exists a W (2n; k), (ii) a complex Hadamard matrix of order 4hc, if there exists an Hadamard matrix of order 4h, (iii) Williamson matrices of order 2cn, if there exist Williamson matrices of order n, (iv) an OD(4cn; 2cs 1 ; \Delta \Delta \Delta ; 2cs u ), if there exists an OD(2n; s 1 ; \Delta \Delta \Delta ; s u ). Also we generalize the above results by using more complex orthogonal matrices. 1 Introduction and Basic Definitions Definition 1 Let C be a (1; \Gamma1; i; \Gammai) matrix of order c satisfying CC = cI, where C is the Hermitian conjugate of C. We call C a complex Hadamard matrix order c. From [6], any complex Hadamard matri..