共查询到20条相似文献,搜索用时 31 毫秒
1.
B.P. Duggal 《Linear algebra and its applications》2008,428(4):1109-1116
A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A∗|2p?|A|2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT∗)?log(T∗T). Let dAB=δAB or ?AB, where δAB∈B(B(H)) is the generalised derivation δAB(X)=AX-XB and ?AB∈B(B(H)) is the elementary operator ?AB(X)=AXB-X. It is proved that if A,B∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (dAB-λ)?1, and dAB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(dAB-λ)Y‖ for all X∈(dAB-λ)-1(0) and Y∈B(H). Furthermore, isolated points of σ(dAB) are simple poles of the resolvent of dAB. A version of the elementary operator E(X)=A1XA2-B1XB2 and perturbations of dAB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for dAB. 相似文献
2.
3.
Shinozaki and Sibuya have shown that the Moore-Penrose inverse (AB)+ can always be expressed as B-A- for generalized inverses A- and B- of matrices A and B, respectively. In this paper, explicit solutions B-mr and A-lr to (AB)+ = B-mrA-lr are given. A class of solutions is obtained which is related to an equation of Greville, and expressions for the general solutions are presented. 相似文献
4.
It is known that if A is positive definite Hermitian, then A·A-1⩾I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A·F(A)⩾AF(A) for all positive definite A if and only if F(A) is a positive multiple of A-1. In addition to the inequality A·A-1⩾I, it is known that A·A-1T⩾I and, stronger, that λmin(A·B)⩾λmin(ABT), for A, B positive definite Hermitian. We also show that λmin(A·B)⩾λmin(AB) and note that λmin(AB) and λmin(ABT) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned. 相似文献
5.
This paper presents conditions which are necessary and sufficient for (AB>)+ = B+Aω for all normalized generalized inverses Aω of the complex matrix A. Corresponding conditions are stated which are equivalent to the situation where (AB)+ = BωA+ is satisfied by each weak generalized inverse Bω of B. The results are applied to theorems by Baskett and Katz and by Schwerdtfeger. 相似文献
6.
《Applied Mathematics Letters》2002,15(6):685-691
A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {rj}j=1p as the r-numbers sequence if r1 = rank(B) and rj = rank[B AB … Aj−1 B] − rank[B AB … Aj−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion Ac of A, such that the pair (Ac, B) has an r-numbers sequence prescribed under some restrictions. 相似文献
7.
Jiri Rohn 《Optimization Letters》2012,6(3):601-603
We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ?1 B, AB ?1, B ?1 A and BA ?1 are P-matrices. A practical method for generating P-matrices is drawn from this result. 相似文献
8.
Daniel A Marcus 《Journal of Combinatorial Theory, Series A》1974,16(3):286-312
Let N be the positive integers; let C be a subset closed under taking divisors; and let A and B be subsets of C such that every member of AB (= {ab: a?A, b?B) is uniquely representable in the form ab, and also AB contains C. Given C, all such pairs (A, B) are found. The result is obtained in a slightly more general setting, and the pairs are replaced by arbitrarily large families. 相似文献
9.
Consider the polynomial tr(A+tB)m in t for positive hermitian matrices A and B with m∈N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB≠0. More precisely, we show—once complex-analytically, once combinatorially—that the k-th coefficient is positive for all integer m?m0, where m0 depends on A, B and k. 相似文献
10.
R.C. Thompson 《Linear algebra and its applications》1976,14(2):135-177
Given n-square Hermitian matrices A,B, let Ai,Bi denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μAi?λBi) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied. 相似文献
11.
Let K ? L be a field extension. Given K-subspaces A, B of L, we study the subspace ?AB? spanned by the product set AB = {ab∣ a ∈ A, b ∈ B}. We obtain some lower bounds on dim K ?AB? and dim K ?B n ? in terms of dim K A, dim K B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson. 相似文献
12.
Vitalij M. Bondarenko Vladimir V. Sergeichuk 《Linear algebra and its applications》2009,430(1):86-105
Pairs (A,B) of mutually annihilating operators AB=BA=0 on a finite dimensional vector space over an algebraically closed field were classified by Gelfand and Ponomarev [Russian Math. Surveys 23 (1968) 1-58] by method of linear relations. The classification of (A,B) over any field was derived by Nazarova, Roiter, Sergeichuk, and Bondarenko [J. Soviet Math. 3 (1975) 636-654] from the classification of finitely generated modules over a dyad of two local Dedekind rings. We give canonical matrices of (A,B) over any field in an explicit form and our proof is constructive: the matrices of (A,B) are sequentially reduced to their canonical form by similarity transformations (A,B)?(S-1AS,S-1BS). 相似文献
13.
Richard Bouldin 《Journal of Mathematical Analysis and Applications》1977,61(2):397-403
Let B be a closed linear transformation of the Banach space X into the Banach space Y and let A be a bounded linear transformation of Y into the Banach space Z. A simple condition is shown to be necessary and sufficient for AB to have closed range. Provided B is relatively regular there is a simple necessary and sufficient condition for AB to be relatively regular. Provided B+ and A+ are pseudoinverses for B and A, respectively, the condition that B+A+ is a pseudoinverse for AB is completely characterized. 相似文献
14.
Morris Newman 《Linear and Multilinear Algebra》2013,61(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
15.
A singular matrix A is perturbed algebraically to obtain a nonsingular matrix B. Particular solutions of Ax=b can be found as unique solutions of Bx=d, where d is an algebraic perturbation of b. More specially, null vectors and generalized null vectors of A can be found as unique solutions of linear systems. It is shown also that B?1AB?1 is a generalized inverse of A. 相似文献
16.
Koenraad M.R. Audenaert 《Linear algebra and its applications》2010,432(1):366-462
We prove an inequality for the spectral radius of products of non-negative matrices conjectured by X. Zhan. We show that for all n×n non-negative matrices A and B, ρ(A°B)?ρ((A°A)(B°B))1/2?ρ(AB), in which ° represents the Hadamard product. 相似文献
17.
Omar Hirzallah 《Journal of Mathematical Analysis and Applications》2003,282(2):578-583
It is shown that if A, B, X are Hilbert space operators such that X?γI, for the positive real number γ, and p,q>1 with 1/p+1/q=1, then |A−B|2?p|A|2+q|B|2 with equality if and only if (1−p)A=B and γ||||A−B|2|||?|||p|A|2X+qX|B|2||| for every unitarily invariant norm. Moreover, if in addition A, B are normal and X is any Hilbert-Schmidt operator, then ‖δA,B2(X)‖2?‖p|A|2X+qX|B|2‖2 with equality if and only if (1−p)AX=XB. 相似文献
18.
Maurice J Dupré 《Journal of Functional Analysis》1974,15(3):244-278
A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p?1(x) is a Hilbert space. However, p?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, Gm(n)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A. 相似文献
19.
Let A be a rectangular matrix of complex numbers whose rows are partitioned into r arbitrary blocks: The Moore-Penrose inverses of each of these blocks are used to form the matrix B = (A1+,…, Ar+). It is shown that 0 ? det (AB) ? 1. This is a generalized version of Hadamard's inequality. 相似文献
20.
Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear b... 相似文献