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In this article, we exhibit under suitable conditions a neat relationship between the least squares g-inverse for a sum of two matrices and the least squares g-inverses of the individual terms. We give a necessary and sufficient condition for the set equations (A?+?B){1,?3}?=?A{1,?3}?+?B{1,?3} and (A?+?B){1,?4}?=?A{1,?4}?+?B{1,?4}.  相似文献   

3.
Secondary symmetric,skewsymmetric and orthogonal matrices   总被引:1,自引:0,他引:1  
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4.
We completely describe the determinants of the sum of orbits of two real skew symmetric matrices, under similarity action of orthogonal group and the special orthogonal group respectively. We also study the Pfaffian case and the complex case.  相似文献   

5.
In this paper, we show that free cumulants can be naturally seen as the limiting value of ``cumulants of matrices'. We define these objects as functions on the symmetric group by some convolution relations involving the generalized moments. We state that some characteristic properties of the free cumulants already hold for these cumulants.  相似文献   

6.
We improve the existence results for holey self-orthogonal Latin squares with symmetric orthogonal mates (HSOLSSOMs) and show that the necessary conditions for the existence of a HSOLSSOM of typeh n are also sufficient with at most 28 pairs (h, n) of possible exceptions. Research supported in part by NSERC Grant A-5320 for the first author, NSF Grants CCR-9504205 and CCR-9357851 for the second author, and NSFC Grant 19231060-2 for the third author.  相似文献   

7.
A matrix C of order n is orthogonal if CCT=dI. In this paper, we restrict the study to orthogonal matrices with a constant m > 1 on the diagonal and ±1's off the diagonal. It is observed that all skew symmetric orthogonal matrices of this type are constructed from skew symmetric Hadamard matrices and vice versa. Some simple necessary conditions for the existence of non-skew orthogonal matrices are derived. Two basic construction techniques for non-skew orthogonal matrices are given. Several families of non-skew orthogonal matrices are constructed by applying the basic techniques to well-known combinatorial objects like balanced incomplete block designs. It is also shown that if m is even and n=0 (mod 4), then an orthogonal matrix must be skew symmetric. The structure of a non-skew orthogonal matrix in the special case of m odd,n=2 (mod 4) and m?1/6n is also studied in detail. Finally, a list of cases with n?50 is given where the existence of non-skew orthogonal matrices are unknown.  相似文献   

8.
In the group of real orthogonal matrices we study the lattice of subgroups containing a group of quasidiagonal orthogonal matrices of fixed type with at most one first-order block.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 94, pp. 73–80, 1979.  相似文献   

9.
For each n≥2, every integral n×n matrix is the sum of at most four squares.  相似文献   

10.
Let ψ be the projectivization (i.e., the set of one-dimensional vector subspaces) of a vector space of dimension ≥ 3 over a field. Let H be a closed (in the pointwise convergence topology) subgroup of the permutation group $\mathfrak{S}_\psi $ of the set ψ. Suppose that H contains the projective group and an arbitrary self-bijection of ψ transforming a triple of collinear points to a non-collinear triple. It is well known from [Kantor W.M., McDonough T.P., On the maximality of PSL(d+1,q), d ≥ 2, J. London Math. Soc., 1974, 8(3), 426] that if ψ is finite then H contains the alternating subgroup $\mathfrak{A}_\psi $ of $\mathfrak{S}_\psi $ . We show in Theorem 3.1 that H = $\mathfrak{S}_\psi $ , if ψ is infinite.  相似文献   

11.
A method of sum composition for construction of orthogona Latin squares was introduced by A. Hedayat and E. Seiden [1]. In this paper we exhibit procedures for constructing a pair of orthogonal Latin squares of size pα + 4 for primes of the form 4m + 1 or p ≡ 1, 2, 4 mod 7. We also show that for any p > 2n and n even one can construct and orthogonal pair of Latin squares of size pα + n using the method of sum composition. We observe that the restriction xy = 1 used by Hedayat and Seiden is sometimes necessary.  相似文献   

12.
Let f be a fourth-degree polynomial over the field of rational numbers Q with leading coefficient 1 which decomposes over Q into the product of two irreducible second-degree polynomials. It is proved that in order that f be the characteristic polynomial of a symmetric matrix with elements in Q, it is necessary and sufficient that all the roots of f be real.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 67, pp. 195–200, 1977.The author expresses his thanks to D. K. Faddeev under whose guidance the present paper was written.  相似文献   

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Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u 2u,u 2u,u 2u − 1) and 2 − (2u 2 + u,u 2,u 2u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.  相似文献   

15.
We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.  相似文献   

16.
If K is a field and char K ≠ 2, then an element α?K is a sum of squares in K if and only if α ? 0 for every ordering of K. This is the famous theorem of Artin and Landau. It has been generalized to symmetric matrices over K by D. Gondard and P. Ribenboim. They have also shown that Artin's theorem on positive definite rational functions has a suitable extension to positive definite matrix functions. In this paper we attain two goals. First, we show that similar theorems are valid for Hermitian matrices instead of symmetric ones. Second, we simplify D. Gondard and P. Ribenboim's proof of their second theorem and strengthen it.  相似文献   

17.
M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 1, pp. 77–78, January–March, 1988.  相似文献   

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Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we extend a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a reminder, and for each such a decomposition we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. Our main tool is the expansion of symmetric functions evaluated on Jucys–Murphy elements.  相似文献   

20.
线性流形上对称正交反对称矩阵反问题的最小二乘解   总被引:1,自引:0,他引:1  
设P是n阶对称正交矩阵,如果n阶矩阵A满足AT=A和(PA)T=-PA,则称A为对称正交反对称矩阵,所有n阶对称正交反对称矩阵的全体记为SARnp.令S={A∈SARnp f(A)=‖AX-B‖=m in,X,B〗∈Rn×m本文讨论了下面两个问题问题Ⅰ给定C∈Rn×p,D∈Rp×p,求A∈S使得CTAC=D问题Ⅱ已知A~∈Rn×n,求A∧∈SE使得‖A~-A∧‖=m inA∈SE‖A~-A‖其中SE是问题Ⅰ的解集合.文中给出了问题Ⅰ有解的充要条件及其通解表达式.进而,指出了集合SE非空时,问题Ⅱ存在唯一解,并给出了解的表达式,从而得到了求解A∧的数值算法.  相似文献   

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