共查询到20条相似文献,搜索用时 31 毫秒
1.
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
2.
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
3.
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
4.
Raphael Loewy 《Linear and Multilinear Algebra》2001,48(4):355-382
Let k and n be positive integers such that k≤n. Let Sn(F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of Sn(F) is said to be a k-subspace if rank A≤k for every AεL.
Now suppose that k is even, and write k=2r. We say a k∥-subspace of Sn(F) is decomposable if there exists in Fn a subspace W of dimension n-r such that xtAx=0 for every xεWAεL.
We show here, under some mild assumptions on kn and F, that every k∥-subspace of Sn(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of Fm,n. 相似文献
Now suppose that k is even, and write k=2r. We say a k∥-subspace of Sn(F) is decomposable if there exists in Fn a subspace W of dimension n-r such that xtAx=0 for every xεWAεL.
We show here, under some mild assumptions on kn and F, that every k∥-subspace of Sn(F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of Fm,n. 相似文献
5.
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
6.
Robert B. Feinberg 《Linear and Multilinear Algebra》1979,7(2):121-122
If A is an n×n matrix over a field F of positive characteristic p, then In=AB-BA, for some matrix B, iff p divides the size of each Jordan block of A. 相似文献
7.
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
8.
Tomasz
uczak 《Discrete Applied Mathematics》1991,30(2-3):213-219
It is shown that for every >0 with the probability tending to 1 as n→∞ a random graph G(n,p) contains induced cycles of all lengths k, 3 ≤ k ≤ (1 − )n log c/c, provided c(n) = (n − 1)p(n)→∞. 相似文献
9.
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
10.
Leroy B. Beasley 《Linear and Multilinear Algebra》1978,6(3):179-183
Let L be a linear transformation on the set of all n×n matrices over an algebraically closed field of characteristic 0. It is shown that if AB=BA implies L(A)L(B)=L(B)L(A) and if either L is nonsingular or the implication in the hypothesis can also be reversed, then L is a sum of a scalar multiple of a similarity transformation and a linear functional times the identity transformation. 相似文献
11.
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
12.
A theorem of the alternatives for the equation Ax + B|x| = b 总被引:4,自引:0,他引:4
Jiri Rohn 《Linear and Multilinear Algebra》2004,52(6):421-426
The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B0|x| = 0 has a nontrivial solution for some matrix B0 of a very special form, |B0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue. 相似文献
13.
Chung-Wei Ha 《Linear and Multilinear Algebra》1988,23(3):263-267
If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]1/n ≥ (det A)1/n + (det B)1/n. 相似文献
14.
15.
Suppose A∈Mn×m(F), B∈Mn×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB). 相似文献
16.
G. T. Rublein 《Linear and Multilinear Algebra》1989,25(4):257-267
Given a pair of n×n matricesA and B, one may form a polynomial P(A,B,λ) which generalizes the characteristic polynomial of BP(B,λ). In particular, when A=I (identity), P(A, B,λ) = P(B,λ), the characteristic polynomial of B. C. Johnson has conjectured [1] (among other things) that when A and B are hermitian and A is positive definite, then P(A,B,λ) has real roots. The case n=2 can be done by hand. In this paper we verify the conjecture for n=3. 相似文献
17.
In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation xn+1=∑i=0kAi/xn−ipi, where k{1,2,…,}, Ai, i{0,1,…,k}, are positive fuzzy numbers, pi, i{0,1,…,k}, are positive constants and xi, i{−k,−k+1,…,0}, are positive fuzzy numbers. 相似文献
18.
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
19.
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
20.
Let Akbe the group of isometries of the space of n-by-n matrices over reals (resp. complexes, quaternions) with respect to the Ky Fan k-norm (see the Introduction for the definitions). Let Γ0 be the group of transformations of this space consisting of all products of left and right multiplications by the elements of SO(n)(resp. U(n), Sp(n)). It is shown that, except for three particular casesAk coincides with the normalizer of Γ in Δ group of isometries of the above matrix space with respect to the standard inner product. We also give an alternative treatment of the case D = Rn = 4k = 2 which was studied in detail by Johnson, Laffey, and Li [4]. 相似文献