共查询到20条相似文献,搜索用时 31 毫秒
1.
Let F be any field and let B a matrix of Fq×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[Aij]i,j∈{1,2}∈F(p+q)×(p+q) with prescribed similarity class and such that A21=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field. 相似文献
2.
John Sinkovic 《Linear algebra and its applications》2010,432(8):2052-411
Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[aij] with aij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G). 相似文献
3.
A matrix A∈Rn×n is called a bisymmetric matrix if its elements ai,j satisfy the properties ai,j=aj,i and ai,j=an-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation problem. 相似文献
4.
Let H be the real quaternion algebra and Hn×m denote the set of all n×m matrices over H. Let P∈Hn×n and Q∈Hm×m be involutions, i.e., P2=I,Q2=I. A matrix A∈Hn×m is said to be (P,Q)-symmetric if A=PAQ. This paper studies the system of linear real quaternion matrix equations
5.
Joe Masaro 《Linear algebra and its applications》2008,429(7):1639-1646
Suppose that Y = (Yi) is a normal random vector with mean Xb and covariance σ2In, where b is a p-dimensional vector (bj), X = (Xij) is an n × p matrix with Xij ∈ {−1, 1}; this corresponds to a factorial design with −1, 1 representing low or high level respectively, or corresponds to a weighing design with −1, 1 representing an object j with weight bj placed on the left and right of a chemical balance respectively. E-optimal designs Z are chosen that are robust in the sense that they remain E-optimal when the covariance of Yi, Yi′ is ρ > 0 for i ≠ i′. Within a smaller class of designs similar results are obtained with respect to a general class of optimality criteria which include the A- and D-criteria. 相似文献
6.
7.
Let f:N→N be a function. Let An=(aij) be the n×n matrix defined by aij=1 if i=f(j) for some i and j and aij=0 otherwise. We describe the Jordan canonical form of the matrix An in terms of the directed graph for which An is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture. 相似文献
8.
We say that a matrix R∈Cn×n is k-involutary if its minimal polynomial is xk-1 for some k?2, so Rk-1=R-1 and the eigenvalues of R are 1,ζ,ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m, A∈Cm×n, S∈Cn×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS-1=ζμA, and A is (R,S,α,μ)-symmetric if RAS-α=ζμA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈Cn×t and B∈Cm×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈Cm×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A. 相似文献
9.
The matrix A = (aij) ∈ Sn is said to lie on a strict undirected graph G if aij = 0 (i ≠ j) whenever (i, j) is not in E(G). If S is skew-symmetric, the isospectral flow maintains the spectrum of A. We consider isospectral flows that maintain a matrix A(t) on a given graph G. We review known results for a graph G that is a (generalised) path, and construct isospectral flows for a (generalised) ring, and a star, and show how a flow may be constructed for a general graph. The analysis may be applied to the isospectral problem for a lumped-mass finite element model of an undamped vibrating system. In that context, it is important that the flow maintain other properties such as irreducibility or positivity, and we discuss whether they are maintained. 相似文献
10.
We consider matrices M with entries mij = m(λi, λj) where λ1, … ,λn are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries wij = 1/m (λi, λj) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means. 相似文献
11.
Wayne Barrett 《Linear algebra and its applications》2011,434(10):2197-2203
Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[ai,j] with ai,j≠0, i≠j if and only if ij is an edge of G. Denote by I↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G1 and G2, then I↗(G)?{(1,1)}=I↗(G1∨K1)∩I↗(G2∨K1)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I↗(G). We use these results to determine I↗(G) for every complete multipartite graph G. 相似文献
12.
Joe Masaro 《Linear algebra and its applications》2008,429(7):1392-1408
Suppose that Y=(Yi) is a normal random vector with mean Xb and covariance σ2In, where b is a p-dimensional vector (bj),X=(Xij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Yi are dependent, i.e., for i≠i′, the covariance of Yi,Yi′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(Xij), where Xij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight bj being weighed on the left and right of a chemical balance respectively. 相似文献
13.
Let F be a field and let m and n be integers with m,n?3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:Mn→Mm that satisfy one of the following conditions:
- 1.
- |F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈Mn and α∈F with ψ(In)≠0.
- 2.
- ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Mn.
14.
15.
Let Mn(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈Mn(C), that is, J=J*,J2=I, let us consider Cn endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cn. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j1,?,jn)=Ir⊕-In-r. For T=(|tik|2)∈Mn(R), the matrices of the form T=(|tik|2jijk), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented. 相似文献
16.
IMA-ISU research group on minimum rank 《Linear algebra and its applications》2010,432(10):2457-2472
The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem. 相似文献
17.
Let F be an infinite field with characteristic not equal to two. For a graph G=(V,E) with V={1,…,n}, let S(G;F) be the set of all symmetric n×n matrices A=[ai,j] over F with ai,j≠0, i≠j if and only if ij∈E. We show that if G is the complement of a partial k -tree and m?k+2, then for all nonsingular symmetric m×m matrices K over F, there exists an m×n matrix U such that UTKU∈S(G;F). As a corollary we obtain that, if k+2?m?n and G is the complement of a partial k-tree, then for any two nonnegative integers p and q with p+q=m, there exists a matrix in S(G;R) with p positive and q negative eigenvalues. 相似文献
18.
Juan-Miguel Gracia 《Linear algebra and its applications》2009,430(4):1196-1215
Given four complex matrices A,B,C and D, where A∈Cn×n and D∈Cm×m, and given a complex number z0: What is the (spectral norm) distance from D to the set of matrices X∈Cm×m such that z0 is a multiple eigenvalue of the matrix
19.
Hong You 《Linear algebra and its applications》2007,426(1):238-254
Let Mn be the space of all n × n complex matrices, and let Γn be the subset of Mn consisting of all n × n k-potent matrices. We denote by Ψn the set of all maps on Mn satisfying A − λB ∈ Γn if and only if ?(A) − λ?(B) ∈ Γn for every A,B ∈ Mn and λ ∈ C. It was shown that ? ∈ Ψn if and only if there exist an invertible matrix P ∈ Mn and c ∈ C with ck−1 = 1 such that either ?(A) = cPAP−1 for every A ∈ Mn, or ?(A) = cPATP−1 for every A ∈ Mn. 相似文献
20.
Mihály Bakonyi 《Integral Equations and Operator Theory》1992,15(2):173-185
H. Dym and I. Gohberg established in [6] necessary and sufficient conditions for the existence and uniqueness of an invertible block matrix F=(Fij)i,j=1,...,n such that Fij=Rij for |i–j|m and F–1 has a band triangular factorization and so (F–1)ij=0 for |i–j|>m. Here Rij, |i–j|m are given block matrices.The aim of this paper is to generalize these results in two directions. First, we shall allow Rij to be an (linear bounded) operator acting between the Hilbert spaces Hj and Hi. Secondly, the set of indices of the given Rij will be more general than banded ones. In fact, we will consider index sets which have an associated graph which is chordal. The case of partial positive operator matrices is also discussed. 相似文献