The generalized Hadamard matrix norms

The concept of quasi-orthogonal matrices ($m$-matrices, minimax matrices) with the quality to have an extremely small value of the maximum element after normalization of their columns or rows is introduced ($m$-norm). Cases to achieve the strict minimum of $m$-norm — Hadamard matrices and a local minimum — generalized Hadamard matrices of odd and even orders are differed. $m$-matrices by the number of their levels — values that take their items are classified. Apart from the Hadamard and Belevich matrices, examples of odd order two- and three-levels matrices Mersenne and Fermat are observed, including even modular duplex Euler matrices replacing matrix Belevich when they do not exist. The formulas to calculate the $M$-matrices and characteristic weights of the right side of their orthogonal columns condition are described. To assess the proximity of $M$-matrices to the Hadamard matrices the weighted $m$-norm ($h$-norm) is proposed, it's equal to the unity for the any Hadamard matrix. Histograms $h$-norms for the family the Hadamard matrices are given. The existence of all Mersenne and Euler matrices for odd and odd orders related 4 are noted. A problem to construct the minimax matrices of Fermat matrix orders is indicated. The structural features and formulas of weighting factors as the basis of alternative definitions of the matrices are noted. Bibliogr. 8. Il. 1.