Some New Results on Circulant Weighing Matrices

We obtain a few structural theorems for circulant weighing matrices whose weight is the square of a prime number. Our results provide new schemes to search for these objects. We also establish the existence status of several previously open cases of circulant weighing matrices. More specifically we show their nonexistence for the parameter pairs (n, k) (here n is the order of the matrix and k its weight) = (147, 49), (125, 25), (200, 25), (55, 25), (95, 25), (133, 49), (195, 25), (11 w, 121) for w < 62.     

On a Class of Symmetric Balanced Generalized Weighing Matrices

Let q be a prime power and m a positive integer. A construction method is given to multiply the parametrs of an -circulant BGW(v=1+q+q ₂+·+q ^m , q ^m , q ^m –q ^m–1) over the cyclic group C _n of order n with (q–1)/n being an even integer, by the parameters of a symmetric BGW(1+q ^m+1, q ^m+1, q ^m+1–q ^m ) with zero diagonal over a cyclic group C _vn to generate a symmetric BGW(1+q+·+q ^2m+1,q ^2m+1,q ^2m+1–q ^2m) with zero diagonal, over the cyclic group C _n . Applications include two new infinite classes of strongly regular graphs with parametersSRG(36(1+25+·+25^2m+1),15(25)^2m+1,6(25)^2m+1,6(25)^2m+1), and SRG(36(1+49+·+49^2m+1),21(49)^2m+1,12(49)^2m+1,12(49)^2m+1).     

Binary Representation of Normalized Symmetric and Correlation Matrices

With a view toward the correlation matrices, it is shown that the normalized real symmetric matrices are the affine hull of the binary correlation matrices, while the convex hull is a proper subset of the correlation matrices. A number of ways to identify the correlation matrices in the affine hull are discussed.     

Binary Representation of Normalized Symmetric and Correlation Matrices

With a view toward the correlation matrices, it is shown that the normalized real symmetric matrices are the affine hull of the binary correlation matrices, while the convex hull is a proper subset of the correlation matrices. A number of ways to identify the correlation matrices in the affine hull are discussed.     

Hadamard Matrices and Dihedral Groups

Let D _2p be a dihedral group of order 2p, where p is an odd integer. Let ZD _2p be the group ring of D _2p over the ring Z of integers. We identify elements of ZD _2p and their matrices of the regular representation of ZD _2p . Recently we characterized the Hadamard matrices of order 28 ([6] and [7]). There are exactly 487 Hadamard matrices of order 28, up to equivalence. In these matrices there exist matrices with some interesting properties. That is, these are constructed by elements of ZD ₆. We discuss relation of ZD _2p and Hadamard matrices of order n=8p+4, and give some examples of Hadamard matrices constructed by dihedral groups.     

Skew-Hadamard Matrices and the Smith Normal Form

We show that the Smith normal form of every skew-Hadamard matrix of order 4m is diag[1,2,...,2, 2m,...,2m,4m]     

Cumulants for Random Matrices as Convolutions on the Symmetric Group,II

In a previous paper we defined some “cumulants of matrices” which naturally converge toward the free cumulants of the limiting non commutative random variables when the size of the matrices tends to infinity. Moreover these cumulants satisfied some of the characteristic properties of cumulants whenever the matrix model was invariant under unitary conjugation. In this paper we present the fitting cumulants for random matrices whose law is invariant under orthogonal conjugation. The symplectic case could be carried out in a similar way.     

Hadamard and Conference Matrices

We discuss new constructions of Hadamard and conference matrices using relative difference sets. We present the first example of a relative -difference set where n – 1 is not a prime power.     

对称双随机矩阵逆特征值问题

主要研究随机矩阵逆特征值问题．特别是对称双随机矩阵和列随机矩阵逆特征值问题．对参考文献[1]与[2]的结论作了一些推广．并给出了—个数值例子．     

有理对称矩阵的准正交相似

对一个有理对称矩阵A,用线性代数的理论直接证明了存在准正交矩阵P使得P~(-1)AP成为A的有理标准形,并给出了P的算法.     

体上两个矩阵乘积的群逆

给出了体上两个矩阵乘积的群逆的存在性的一个等价条件及若干充分条件。     

Construction of Weighing Matrices W(17, 9) Having the Intersection Number 8

It is unknown( [1], [2], [4] ) whether there exists a weighing matrix W(17,9) or not. The intersection number of W(17,9) is 6 or 8. We completed the classification of weighing matrices W(17, 9) having the intersection number 8. These matrices are classified into 925 non-equivalent classes. We shall show eight representative weighing matrices only together with their automorphism groups and g-distributions. Of these eight matrices, two have the same g-distribution.     

A Recursive Construction for New Symmetric Designs

We introduce a recursive construction of regular Handamard matrices with row sum 2h for h=±3ⁿ. Whenever q=(2h – 1)² is a prime power, we construct, for every positive integer m, a symmetric designs with parameters (4h²(q^m+1 – 1)/(q – 1), (2h² – h)q^m, (h² – h)q^m).     

Williamson Matrices and a Conjecture of Ito's

We point out an interesting connection between Williamson matrices and relative difference sets in nonabelian groups. As a consequence, we are able to show that there are relative (4t, 2, 4t, 2t)-difference sets in the dicyclic groups Q _8t = a, b|a ^4t = b ⁴ = 1, a ^2t = b ², b ^-1ab = a^-1 for all t of the form t = 2^a · 10 ^b · 26 ^c · m with a, b, c 0, m 1\ (mod 2), whenever 2m-1 or 4m-1 is a prime power or there is a Williamson matrix over _m. This gives further support to an important conjecture of Ito IT5 which asserts that there are relative (4t, 2, 4t, 2t)-difference sets in Q _8t for every positive integer t. We also give simpler alternative constructions for relative (4t, 2, 4t, 2t)-difference sets in Q _8t for all t such that 2t - 1 or 4t - 1 is a prime power. Relative difference sets in Q _8t with these parameters had previously been obtained by Ito IT1. Finally, we verify Ito's conjecture for all t 46.     

Remarks on Hadamard Matrices and Lattices

Automorphism groups of Hadamard matrices are investigated from the point of view of integral representation theory. Interesting examples involving the Mathieu groups M ₁₂ and M ₂₄ and the Leech lattice are dicussed.     

Spectral Decomposition of Symmetric Operator Matrices

The authors study symmetric operator matrices in the product of Hilbert spaces H = H₁×H₂, where the entries are not necessarily bounded operators. Under suitable assumptions the closure L_o exists and is a selfadjoint operator in H. With L_o, the closure of the transfer function is considered. Under the assumption that there exists a real number β < inf p(A) such that M(β)<< 0, it follows that β ε p(L_o). Applying a factorization result of A.I. Virozub and V.I. Matsaev [VM] to the holomorphic operator function M(λ, the_spectral subspaces of L_o corresponding to the intervals ] — ∞, β] and [β, ∞[ and the restrictions of L_o to these subspaces are characterized. Similar results are proved for operator matrices which are symmetric in a Krein space.     

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices. A. Massey’s current address: Department of Mathematics, UCLA, Los Angeles, CA 90095, USA. e-mail: amassey3102@math.ucla.edu.     

The Eigenvalues of Very Sparse Random Symmetric Matrices

Let W _n be an n × n random symmetric sparse matrix with independent identically distributed entries such that the values 1 and 0 are taken with probabilities p/n and 1-p/n, respectively; here is independent of n. We show that the limit of the expected spectral distribution functions of W _n has a discrete part. Moreover, the set of positive probability points is dense in (- +). In particular, the points , and 0 belong to this set.