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1.
Mieko Yamada 《Graphs and Combinatorics》1988,4(1):297-301
In this paper we give a new series of Hadamard matrices of order 2
t
. When the order is 16, Hadamard matrices obtained here belong to class II, class V or to class IV of Hall's classification [3]. By combining our matrices with the matrices belonging to class I, class II or class III obtained before, we can say that we have direct construction, namely without resorting to block designs, for all classes of Hadamard matrices of order 16.Furthermore we show that the maximal excess of Hadamard matrices of order 22t
is 23t
, which was proved by J. Hammer, R. Levingston and J. Seberry [4]. We believe that our matrices are inequivalent to the matrices used by the above authors. More generally, if there is an Hadamard matrix of order 4n
2 with the maximal excess 8n
3, then there exist more than one inequivalent Hadamard matrices of order 22t
n
2 with the maximal excess 23t
n
3 for anyt 2. 相似文献
2.
Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UXtV for all matrices X where Xt denotes the transpose of X. 相似文献
3.
P. V. Omel’chenko 《Ukrainian Mathematical Journal》2009,61(10):1578-1588
We study the problem of the reduction of self-adjoint block matrices B = (B
ij
) with given graph by a group of unitary block diagonal matrices. Under the condition that the matrices B
2 and B
4 are orthoscalar, we describe the graphs of block matrices for which this problem is a problem of *-finite, *-tame, or *-wild
representation type. 相似文献
4.
This article derives from first principles a definition of equivalence for higher‐dimensional Hadamard matrices and thereby a definition of the automorphism group for higher‐dimensional Hadamard matrices. Our procedure is quite general and could be applied to other kinds of designs for which there are no established definitions for equivalence or automorphism. Given a two‐dimensional Hadamard matrix H of order ν, there is a Product Construction which gives an order ν proper n‐dimensional Hadamard matrix P(n)(H). We apply our ideas to the matrices P(n)(H). We prove that there is a constant c > 1 such that any Hadamard matrix H of order ν > 2 gives rise via the Product Construction to cν inequivalent proper three‐dimensional Hadamard matrices of order ν. This corrects an erroneous assertion made in the literature that ”P(n)(H) is equivalent to “P(n)(H′) whenever H is equivalent to H′.” We also show how the automorphism group of P(n)(H) depends on the structure of the automorphism group of H. As an application of the above ideas, we determine the automorphism group of P(n)(Hk) when Hk is a Sylvester Hadamard matrix of order 2k. For ν = 4, we exhibit three distinct families of inequivalent Product Construction matrices P(n)(H) where H is equivalent to H2. These matrices each have large but non‐isomorphic automorphism groups. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 507–544, 2008 相似文献
5.
Marjeta Kramar 《Linear and Multilinear Algebra》2013,61(1):13-25
We describe the structure of irreducible matrix groups with submultiplicative spectrum. Since all such groups are nilpotent, the study is focused on p-groups. We obtain a block-monomial structure of matrices in irreducible p-groups and build polycyclic series arising from that structure. We give an upper bound to the exponent of these groups. We determine all minimal irreducible groups of p× p matrices with submultiplicative spectrum and discuss the case of p 2× p 2 matrices if p is an odd prime. 相似文献
6.
We present a new bound for suprema of a special type of chaos process indexed by a set of matrices, which is based on a chaining method. As applications we show significantly improved estimates for the restricted isometry constants of partial random circulant matrices and time‐frequency structured random matrices. In both cases the required condition on the number m of rows in terms of the sparsity s and the vector length n is m ? s log2 s log2 n. © 2014 Wiley Periodicals, Inc. 相似文献
7.
A weighing matrix of order n and weight m2 is a square matrix M of order n with entries from {-1,0,+1} such that MMT=m2I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.MT=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2pr and weight p2 where p is a prime and p≥ 5.Communicated by: K.T. Arasu 相似文献
8.
Karlheinz Gröchenig Ziemowit Rzeszotnik Thomas Strohmer 《Integral Equations and Operator Theory》2010,67(2):183-202
The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based
on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted
ℓ
p
-spaces. In particular, the stability of the finite section method on ℓ
2 implies its stability on weighted ℓ
p
-spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore,
we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable
to large classes of unstructured non-hermitian matrices as well as to least squares problems. 相似文献
9.
Dana Randall 《Random Structures and Algorithms》1993,4(1):111-118
We present an efficient algorithm for generating an n × n nonsingular matrix uniformly over a finite field. This algorithm is useful for several cryptographic and checking applications. Over GF[2] our algorithm runs in expected time M(n) + O(n2), where M(n) is the time needed to multiply two n × n matrices, and the expected number of random bits it uses is n2 + 3. (Over other finite fields we use n2 + O(1) random field elements on average.) This is more efficient than the standard method for solving this problem, both in terms of expected running time and the expected number of random bits used. The standard method is to generate random n × n matrices until we produce one with nonzero determinant. In contrast, our technique directly produces a random matrix guaranteed to have nonzero determinant. We also introduce efficient algorithms for related problems such as uniformly generating singular matrices or matrices with fixed determinant. © 1993 John Wiley & Sons, Inc. 相似文献
10.
If M is any complex matrix with rank (M + M * + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + MT + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented. 相似文献
11.
Some examples of orthogonal matrix polynomials satisfying odd order differential equations 总被引:2,自引:1,他引:1
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form
W(t)=tαe-teAttBtB*eA*t,