共查询到20条相似文献,搜索用时 15 毫秒
1.
We investigate the structure of the multiplicative semigroup generated by the set of matrices that are unitarily equivalent to a given singular matrix A. In particular, we give necessary and sufficient conditions, in terms of the singular values of A, for such a semigroup to consist of all matrices of rank not exceeding the rank of A. 相似文献
2.
In this paper we study the possible spectra among matrices congruent to a given AεMn(C). It is important to distinguish singular A from nonsingular and, among non-singular matrices, to distinguish where 0 lies relative to the field of values of A. 相似文献
3.
The column rank of an m by n matrix A over max algebra is the weak dimension of the column space of A. We compare the column rank with rank of matrices over max algebra. We also characterize the linear operators which preserve the column rank of matrices over max algebra. 相似文献
4.
Yongge Tian 《Linear and Multilinear Algebra》2002,49(4):269-288
A matrix X is called an outer inverse for a matrix A if XAX=X. In this paper, we present some basic rank equalities for difference and sum of outer inverses of a matrix, and apply them to characterize various equalities related to outer inverses, Moore-Penrose inverses, group inverses, Drazin inverses and weighted Moore-Penrose inverses of matrices. 相似文献
5.
In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A*) and (B, B*) to coincide, but only to be close. 相似文献
6.
Morris Newman 《Linear and Multilinear Algebra》1978,5(4):265-266
Let R be a principal ideal ringRn the ring of n × n it matrices over R. It is shown that if A, B, X, Y are elements of R* such that A = XB, B = YA, then A and B are left equivalent. Some consequences are given. 相似文献
7.
A commutative ring A has quadratic stable range 1 (qsr(A) = 1) if each primitive binary quadratic form over A represents a unit. It is shown that qsr(A) = 1 implies that every primitive quadratic form over A represents a unit, A has stable range 1 and finitely generated constant rank projectives over A are free. A classification of quadratic forms is provided over Bezout domains with characteristic other than 2, quadratic stable range 1, and a strong approximation property for a certain subset of their maximum spectrum. These domains include rings of holomorphic functions on connected noncompact Riemann surfaces. Examples of localizations of rings of algebraic integers are provided to show that the classical concept of stable range does not behave well in either direction under finite integral extensions and that qsr(A) = 1 does not descend from such extensions. 相似文献
8.
We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions. 相似文献
9.
Glenn Merlet 《Linear algebra and its applications》2010,432(8):1923-1757
We investigate the action of semigroups of d×d matrices with entries in the max-plus semifield on the max-plus projective space. Recall that semigroups generated by one element with projectively bounded image are projectively finite and thus contain idempotent elements.In terms of orbits, our main result states that the image of a minimal orbit by an idempotent element of the semigroup with minimal rank has at most d! elements. Moreover, each idempotent element with minimal rank maps at least one orbit onto a singleton.This allows us to deduce the central limit theorem for stochastic recurrent sequences driven by independent random matrices that take countably many values, as soon as the semigroup generated by the values contains an element with projectively bounded image. 相似文献
10.
Morris Newman 《Linear and Multilinear Algebra》1975,3(1):95-98
The principal results are that if A is an integral matrix such that AAT is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic. 相似文献
11.
12.
Claire Cates Haulk John Drew Charles R. Johnson J. Hunter Tart 《Linear and Multilinear Algebra》1997,43(1):35-51
For a given n-by-n matrix A, we consider the set of matrices which commute with A and all of whose principal submatrices commute with the corresponding principal submatrices of A. The properties of this set are examined, with particular attention to its dimension. 相似文献
13.
J. R. Wall 《Linear and Multilinear Algebra》1975,2(4):337-340
For a doubly stochastic matrix A, each of the equations x:aty= A and X Aty=t is shown to have doubly stochastic solutions X and Y if and only if A lies in a subgroup of the semigroup of all doubly stochastic matrices of a given order. All elements of this semigroup which are left regular, right regular, or intra-regular are identified. 相似文献
14.
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
15.
Inequalities that compare unitarily invariant norms of A - B and those of AΓ - ΓB and Γ-1A - B Γ-1 are obtained, where both A and B are either Hermitian or unitary or normal operators and Γ is a positive definite operator in a complex separable Hilbert space. These inequalities are then applied to derive bounds for spectral variation of diagonalisable matrices. Our new bounds improve substantially previously published bounds. 相似文献
16.
Zero-term rank preservers 总被引:2,自引:0,他引:2
We obtain characterizations of those linear operators that preserve zero-term rank on the m×n matrices over antinegative semirings. That is, a linear operator T preserves zero-term rank if and only if it has the form T(X)=P(B∘X)Q, where P, Q are permutation matrices and B∘X is the Schur product with B whose entries are all nonzero and not zero-divisors. 相似文献
17.
Fernando C. Silva 《Linear and Multilinear Algebra》1988,24(1):51-58
Let F be a field and let A,B be n × n matrices over I. We study the rank of A' - B' when A and B run over the set of matrices similar to A and B, respectively. 相似文献
18.
Let Mn be the set of n×n matrices and r a nonnegative integer with r ≤ n. It is known,from Lie groups, that the rank r idempotent matrices in Mn form an arcwise connected 2n (n-r)-dimensional analytic manifold. This paper provides an elementary proof of this result making it accessible to a larger audience. 相似文献
19.
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
20.
Linear operators that preserve pairs of matrices which satisfy extreme rank properties 总被引:4,自引:0,他引:4
LeRoy B. Beasley Sang-Gu Lee Seok-Zun Song 《Linear algebra and its applications》2002,350(1-3):263-272
A pair of m×n matrices (A,B) is called rank-sum-maximal if rank(A+B)=rank(A)+rank(B), and rank-sum-minimal if rank(A+B)=|rank(A)−rank(B)|. We characterize the linear operators that preserve the set of rank-sum-minimal matrix pairs, and the linear operators that preserve the set of rank-sum-maximal matrix pairs over any field with at least min(m,n)+2 elements and of characteristic not 2. 相似文献