共查询到20条相似文献,搜索用时 31 毫秒
1.
Raphael Loewy 《Linear and Multilinear Algebra》1993,36(2):115-123
We prove the following result. Let F be an infinite field of characteristic other than two. Let k be a positive integer. Let Sn(F) denote the space of all n × n symmetric matrices with entries in F, and let T:Sn(F)→Sn(F) be a linear operator. Suppose that T is rank-k nonincreasing and its image contains a matrix with rank higher than K. Then, there exist λεF and PεFn,n such that T(A)=λPAPt for all AεSn(F). λ can be chosen to be 1 if F is algebraically closed and ±1 if F=R, the real field. 相似文献
2.
Roy Meshulam 《Linear and Multilinear Algebra》1990,26(1):39-41
It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1. 相似文献
3.
Carlos Gamas 《Linear and Multilinear Algebra》2000,47(2):151-173
Let λ be an irreducible character of Sn corresponding to the partition (r,s) of n. Let A be a positive semidefinite Hermitian n × n matrix. Let dλ(A) and per(A) be the immanants corresponding to λ and to the trivial character of Sn, respectively. A proof of the inequality dλ(A)≤λ(id)per(A) is given. 相似文献
4.
Morris Newman 《Linear and Multilinear Algebra》1982,11(4):363-366
Let Rbe a principal ideal ringRn the ring of n× nmatrices over R, and dk(A) the kth determinantal divisor of Afor 1 ≤ k≤ n, where Ais any element of Rn, It is shown that if A,BεRn, det(A) det(B:) ≠ 0, then dk(AB) ≡ 0 mod dk(A) dk(B). If in addition (det(A), det(B)) = 1, then it is also shown that dk(AB) = dk(A) dk(B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants. 相似文献
5.
Inertially arbitrary patterns 总被引:11,自引:0,他引:11
An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (rst) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n≥2) skew-symmetric sign pattern Sn with each upper off-diagonal entry positive, the (1,1) entry negative, the (nn) entry positive, and every other diagonal entry zero. We prove that Sn is an IAP. 相似文献
6.
Chi-Kwong Li Ilya Spitkovsky Sudheer Shukla 《Linear algebra and its applications》1998,270(1-3):323-349
Let Mn be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A Mn is defined by Wk(A) = (1/k)∑jk=1xj*Axj : x1, …, xk is an orthonormal set in n]. It is known that tr A/n = Wn(A) Wn−1(A) W1(A). We study the condition on A under which Wm(A) = Wk(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range. 相似文献
7.
We consider scalar-valued matrix functions for n×n matrices A=(aij) defined by Where G is a subgroup of Sn the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.
With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1). 相似文献
With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1). 相似文献
8.
We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A1,A2, an n × n nilpotent Toeplitz matrix Nn, and an n × n cyclic permutation matrix Sn(s) such that the numbers of flat portions on the boundaries of W(A1⊕Nn) and W(A2⊕Sn(s)) are, respectively, 2(n - 2) and 2n. 相似文献
9.
It is proved that computing the subordinate matrix norm ∥A∥∞1 is NP-hard, Even more, existence of a polynomial-time algorithm for computing this norm with relative accuracy less than 1/(4n2), where n is matrix size, implies P = NP. 相似文献
10.
Charles R. Johnson 《Linear and Multilinear Algebra》1990,26(1):5-8
Let a positive definite Hermitian matrix HεMn(C) be decomposed as H=A + iB, with A, B ε Mnm(R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A. 相似文献
11.
Huang Liping 《Linear and Multilinear Algebra》1998,45(2):99-107
Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial qA(λ) of A over F. By using qA(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix. 相似文献
12.
Thomas H. Pate 《Linear and Multilinear Algebra》2003,51(3):263-278
If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible. 相似文献
13.
D. B. Hunter 《Linear and Multilinear Algebra》1983,13(4):357-366
A method is described for constructing in an explicit form an irreducible representation T of Mn(F), the set of all n × n matrices over the real or complex field F, satisfying the condition T(A*)=T*(A) for all A∈Mn(F). 相似文献
14.
Fernando C. Silva 《Linear and Multilinear Algebra》1990,27(4):317-323
Let F be a field and let A and n × n matrices over F. We study some properties of A' + B' and A'B', when A' and B' run over the sets of the matrices similar to A and B, respectively. 相似文献
15.
Let U⊗V denote the tensor product of two finite dimensional vector spaces U and V over an infinite field. Let k be a positive integer such that k≤dim U and k≤ dim V Let Dk denote the set of all non-zero elements of U⊗V of rank less than k. In this paper we study linear transformations T on U⊗V such that (TDk)⊆Dk. 相似文献
16.
Products of involutory matrices. I 总被引:1,自引:0,他引:1
C. S. Ballantine 《Linear and Multilinear Algebra》1977,5(1):53-62
It is shown that, for every integer ≥1 and every field F, each n×n matrix over F of determinant ±1 is the product of four involutory matrices over F. Products of three ×n involutory matrices over F are characterized for the special cases where n≤4 or F has prime order ≤5. It is also shown for every field F that every matrix over F of determinant ±1 having no more than two nontrivial invariant factors is a product of three involutory matrices over F. 相似文献
17.
Let Knbe the convex set of n×npositive semidefinite doubly stochastic matrices. If Aε kn, the graph of A,G(A), is the graph on n vertices with (i,j) an edge if aij ≠ 0i≠ j. We are concerned with the extreme points in Kn. In many cases, the rank of Aand G(A) are enough to determine whether A is extreme in Kn. This is true, in particular, if G(A)is a special kind of nonchordal graph, i.e., if no two cycles in G(A)have a common edge. 相似文献
18.
Xian Zhang 《Linear and Multilinear Algebra》2004,52(5):349-358
Suppose F is a field of characteristic not 2. Let MnF and SnF be the n × n full matrix space and symmetric matrix space over F, respectively. All additive maps from SnF to SnF (respectively, MnF) preserving Moore-Penrose inverses of matrices are characterized. We first characterize all additive Moore-Penrose inverse preserving maps from SnF to MnF, and thereby, all additive Moore-Penrose inverse preserving maps from SnF to itself are characterized by restricting the range of these additive maps into the symmetric matrix space. 相似文献
19.
T. H. Pate 《Linear and Multilinear Algebra》1981,10(2):103-105
Let A be an nk × nk positive semi-definite symmetric matrix partitioned into blocks Aij each of which is an n × n matrix. In [2] Mine states a conjecture of Marcus that per(A) ≥ per(G) where G is the k × k matrix [per(Aij)]. In this paper we prove a weaker inequality namely that per(A) ≥ (k!)-1per(G). 相似文献
20.
C. S. Ballantine 《Linear and Multilinear Algebra》1975,3(1):19-23
Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC-1for some C = C-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger. 相似文献