Hadamard matrix family catalogue and on-line algorithms

© Nickolay A. Balonin and Jennifer Seberry, 23.10.2014

Definition 1: Quasi-orthogonal matrix is called regular, if the sums of the elements in each row or column are equal to each other. These are also called regular Cretan matrices.


This regular structure M11 has no equality L*(v–1)=10, k*(k–1)=12, so orthogonal version has 6 levels
(system 4*b*b-6*a*b+4*a*a=0; 2*b*b-4*a*b+5*a*a=0; has solution a=0, b=0)!


In constrast to regular Hadamard matrices, regular Cretan matrices are widespread, because they have varying level, levels or a linear combination of levels and they provide an opportunity for simple regular structures, such as circulant matrices, to have sums of column or row entries which are invariant for each of them.

Regular Cretan matrices can be constructed using matrices R(b) of regular structure which have sums of its columns and rows, independently, on varying levels –b equal to each other. If that is not possible, R(b) can be used also to construct non regular quasi-orthogonal matrix of the next order n+1:


Equation R(b)e=γe, where γ is an eigen value of the matrix R(b), e=[σ, σ, σ .. σ]T is its eigen vector consisting of some constant border entries, leads to the necessary condition A(b)TA(b) is a diagonal matrix for many such cases.

Example 1. Circulant matrix R(b) with 8 entries {–a,a,–b,–b,–a,–b,a,a} gives the following quasi-orthogonal matrix order 9 with parameters a=1, b=0.4641, γ=0.3923, σ=0.8859.

Maximum-determinant matrix A9 and histogram of its entries

In this original case we have all parameters which are all different from each other: the parameters of the core R(b) and the border, giving five levels. This Cretan matrix of quasi-orthogonal matrix has maximal determinant: it is one of 5 BM-matrices.

We have some less extremal case, when R(b) is a two-level a, –b matrix, resolvable as quasi-orthogonal itself. The regular Hadamard matrices are a source of Cretan RHM or quasi-orthogonal matrices, they are known for orders equal to Fermat numbers and many other orders 4t+1.

Fermat matrix F17 and histogram of its entries

There are circulant or more complicated cores of symmetric conference matrices, i.e. weighing matrices of order n with weight n–1 for example when n–1 is prime. They can be used, changing two parameters of R(b,δ) of order v=n–1: b and variable diagonal entry δ, to reach the ballance γ=δ=0, σ=b=1. Using the set of known conference matrices, this method gives a solution for the reversed task: to find quasi-orthogonal matrix R(b,δ), choosing the linear combination b=1–2δ. In this case δ=1/(1+sqrt(v)) and v=4t+1.

Conference matrix C6 and regular quasi-orthogonal matrix R5

Two level matrices, described by the content of the product R(b)TR(b)=ω(b)I+ε(b)J, where where I is the identity matrix and J the matrix of all ones, have been studied for regular structures, known as Cretan-SBIBD. The equation ε(b)=0 gives the necessary value of level –b. This task can be solved for some odd orders.

Regular Cretan matrix, CM-SBIBD based on the SBIBD(31,6,1), and new CM(32) with border

The solution ε(b)=0 gives also the well known set of Hadamard matrices based on the regular cores orders 3, 7, 11, and so on.

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