Almost Hadamard Matrices: General Theory and Examples

We develop a general theory of almost Hadamard matrices. These are by definition the matrices H is an element of M-N(R) having the property that U = H/root N is orthogonal, and is a local maximum of the 1-norm on O(N). Our study includes a detailed discussion of the circulant case (H-ij = gamma(j-i)) and of the two-entry case (H-ij is an element of {x, y}), with the construction of several families of examples, and some 1-norm computations.