MERSENNE, EULER, SEIDEL, FERMAT LEVELS Hadamard matrix family catalogue and on-line algorithms

© Nickolay A. Balonin, 1.05.2014

TABLE OF LEVEL FUNCTIONS

There are 5 main families (Mersenne, Euler, Hadamard, Fermat, Seidel) of absolute and local maximum determinat matrices: |det(X)|→max; |xij|≤1; X'X=ω(n)I.

Local maximum determinant matrices ,  can be described by functions of their levels. Varying levels a=1, –b; 0≤b≤1 (can be irrational) allow to construct such simple structures (imposible with integer levels for some orders) as circulant and two-circulant matrices.

Definition 1. The values of the entries of a matrix are called levels .

Hadamard matrices are optimal two-level matrices [2, 3], symmetric conference matrices  and weighing matrices are three-level matrices . Quasi-orthogonal matrices with maximal determinant of odd orders have been discovered with a larger number of levels .

Definition 2. Quasi-orthogonal matrix, X, of order n, satisfies XTX = ω(n)I, where ω(n)≤n is the weight, and defined by a function ω(n) or functions a(n)=1, b(n), .. of its levels.

We will denote the levels of two-level matrices as a, –b; for positive 0 ≤ ba, a = 1.

Determinant |det(X)|=ω(n)n/2, ω(n)=nψ(n)2, where ψ – a normalized determinant, ψ(n)=1 for Hadamard matrices.

 ω(n)=nψ(n)2

Hadamard inequality |det(X)|≤nn/2 gives an estimation of determinant, while |det(X)|=nn/2×ψ(n)n gives a possible value.

MAIN MATRIX FAMILIES  Hadamard matrices H12 and H16

 Hadamard matrix. A quasi-orthogonal matrix, named Hadamard matrix  H, is a two-level quasi-orthogonal matrix of order n with level functions a=1,–b; b=1. Hadamard conjecture : Hadamard matrices exist for n=1,2 and 4t, t is integer. Their orders cover n = 2k, k is integer, for so called elementary Hadamard matrices set.Normalized determinant ψ = 1.    Mersenne matrices M11 and M15

 Mersenne matrix. A quasi-orthogonal matrix, named Mersenne matrix M , is a two-level quasi-orthogonal matrix of order n with level functions a=1, –b; where b=t/(t+sqrt(t)), t=(n+1)/4. Conjecture [8,12,15]: Mersenne matrices exist for n=4t–1, t is integer.Their orders cover Mersenne numbers n = 2k–1, k is integer, for so called elementary Mersenne matrices set (or pure Mersenne matrices).Invariant of Mersenne matrices is difference 1 between numbers of positive and negative entries in every column (row), so weight ω = (n+1)/2+(n–1)b2/2=2t+(2t–1)b2.Normalized determinant ψ = sqrt((2t+(2t–1)b2))/(4t–1).  Euler matrices E10 and E14

 Euler matrix. A quasi-orthogonal matrix, named Euler matrix E , is a four-level quasi-orthogonal matrix of order n, it can be observed as two-circulant (or two blocks A, B) matrices with two level functions a=1, –b; where b=t/(t+sqrt(2t)), t=(n+2)/4. Conjecture [8,12,15]: Euler matrices exist for n=4t–2, t is integer.Their orders cover singular orders for the symmetric conference matrices: the latests do not exist if n–1 is not sum of two squares. Invariant of Euler matrices is difference 2 between numbers of ±1 and ±b entries in every column (row), so weight ω = (n+2)/2+(n–2)b2/2=2t+2t–1)b2.Normalized determinant ψ = sqrt((2t+2(t–1)b2))/(4t–2).  Seidel matrices S9 and S13

 Seidel matrix. A quasi-orthogonal matrix, named Seidel matrix S , is a quasi-orthogonal matrix of order n with level functions a=1, –b, d; where b=1–2d, d=1/(1+sqrt(n)) belongs to diagonal entries, t=(n+3)/4. Their orders cover singular points of symmetric conference matrices; Seidel matrices do not exist if n is not sum of two squares . Invariant of Seidel matrices is difference 1 between numbers of positive and negative entries in every column (row), so weight ω = (n–1)(1+b2)/2+d2=2(t–1)(1+b2)+d2. Normalized determinant ψ = sqrt((2(t–1)(1+b2)+d2)/(4t–3).  Hadamard H16 anf Fermat F17

 Fermat matrix. A quasi-orthogonal matrix, named Fermat matrix F , is a quasi-orthogonal matrix of order n with level functions a=1, –b, s; for n=3, 2a=b=s=1; for n>3, q=n–1=4u2, p=q+sqrt(q), we have b≤s

MERSENNE MATRIX LEVEL b

b=(q–(4q)1/2)/(q–4), q=n+1, n=4k–1.

EULER MATRIX LEVEL b

b=(q–(8q)1/2)/(q–8), q=n+2, n=4k–2.

DETERMINANT GRAPH D=1/mn=nn/2/hn

Hadamard H, Belevitch C, Euler E, Mersenne M, Fermat F matrices, Cyan is Proclus.

References

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