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Almost Hadamard matrices: general theory and examples. (English) Zbl 1263.15030

A Hadamard matrix is a square matrix with \(\mp1\) entries the rows of which are pairwise orthogonal and an almost Hadamard matrix is defined as a square matrix \(H\in M_{N}(R)\) such that \(U=H/\sqrt{N}\) is orthogonal and is a local maximum of the \(1\)-norm on \(O(N)\). A matrix \(H\in M_{N}(\mathbb{C})\) is called circulant if \(M_{ij}\) depends only on \(j-i\), taken modulo \(N\). In this paper, the authors try to provide a systematic study of such matrices, with the construction of a number of nontrivial examples, and with the development of some general theory. First, the authors study the almost Hadamard matrices which are circulant. Then they give some theorems and construct several examples about these matrices. Later, they deal with the almost Hadamard matrices having only two entries, \(H\in M_{N}(x,y)\) with \(x,y\in R\). Then they also construct some theorems and give several examples about these matrices. Moreover, the authors give a detailed discussion on some \(1\)-norm computations.

MSC:

15B34 Boolean and Hadamard matrices
15B05 Toeplitz, Cauchy, and related matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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