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1.
Let TRn×n be an irreducible stochastic matrix with stationary distribution vector π. Set A = I − T, and define the quantity , where Aj, j = 1, … , n, are the (n − 1) × (n − 1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ3 provides a sensitive measure of the conditioning of π under perturbation of T. Moreover, it is known that .In this paper, we investigate the class of irreducible stochastic matrices T of order n such that , for such matrices correspond to Markov chains with desirable conditioning properties. We identify some restrictions on the zero-nonzero patterns of such matrices, and construct several infinite classes of matrices for which κ3 is as small as possible.  相似文献   

2.
The paper studies the eigenvalue distribution of some special matrices. Tong in Theorem 1.2 of [Wen-ting Tong, On the distribution of eigenvalues of some matrices, Acta Math. Sinica (China), 20 (4) (1977) 273-275] gives conditions for an n × n matrix A ∈ SDn ∪ IDn to have |JR+(A)| eigenvalues with positive real part, and |JR-(A)| eigenvalues with negative real part. A counter-example is given in this paper to show that the conditions of the theorem are not true. A corrected condition is then proposed under which the conclusion of the theorem holds. Then the corrected condition is applied to establish some results about the eigenvalue distribution of the Schur complements of H-matrices with complex diagonal entries. Several conditions on the n × n matrix A and the subset α ⊆ N = {1, 2, … , n} are presented such that the Schur complement matrix A/α of the matrix A has eigenvalues with positive real part and eigenvalues with negative real part.  相似文献   

3.
Let F be a field with ∣F∣ > 2 and Tn(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A1, …, Ak ∈ Tn(F) is called rank reverse permutable if rank(A1 A2 ? Ak) = rank(Ak Ak−1 ? A1). We characterize the linear maps on Tn(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.  相似文献   

4.
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   

5.
6.
Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A = (A1, … , An) and B = (B1, … , Bn) of elements in A, we define the elementary operator RA,B on A by the relation for all X in A. For a single operator AA, we define the two particular elementary operators LA and RA on A by LA(X) = AX and RA(X) = XA, for every X in A. We denote by d(RA,B) the supremum of the norm of RA,B(X) over all unit rank one operators on E. In this note, we shall characterize: (i) the supremun d(RA,B), (ii) the relation , (iii) the relation d(LA − RB) = ∥A∥ + ∥B∥, (iv) the relation d(LARB − LBRA) = 2∥A∥ + ∥B∥. Moreover, we shall show the lower estimate d(LA − RB) ? max{supλV(B)A − λI∥, supλV(A)B − λI∥} (where V(X) is the algebraic numerical range of X in A).  相似文献   

7.
Let R ∈ Cn×n be a nontrivial involution, i.e., R2 = I and R ≠ ±I. A matrix A ∈ Cn×n is called R-skew symmetric if RAR = −A. The least-squares solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are firstly derived, then the solvability conditions and the solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are given. The solutions of the corresponding optimal approximation problem with R∗ = R for R-skew symmetric matrices are also derived. At last an algorithm for the optimal approximation problem is given. It can be seen that we extend our previous results [G.X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. 189 (2007) 482-489] and the results proposed by Zhou et al. [F.Z. Zhou, L. Zhang, X.Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589].  相似文献   

8.
9.
10.
Let σ = (λ1, … , λn) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ1, a diagonal entry c and let τ = (μ1, … , μm) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ1. We show how to construct a nonnegative symmetric matrix C with the spectrum
(λ1+max{0,μ1-c},λ2,…,λn,μ2,…,μm).  相似文献   

11.
12.
A collection A1A2, …, Ak of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A1A2, …, Ak. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A1, …, Ak when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)  相似文献   

13.
Let ∥ · ∥ be the Frobenius norm of matrices. We consider (I) the set SE of symmetric and generalized centro-symmetric real n × n matrices Rn with some given eigenpairs (λjqj) (j = 1, 2, … , m) and (II) the element in SE which minimizes for a given real matrix R. Necessary and sufficient conditions for SE to be nonempty are presented. A general form of elements in SE is given and an explicit expression of the minimizer is derived. Finally, a numerical example is reported.  相似文献   

14.
Let Mn be the algebra of all n×n complex matrices and Γn the set of all k-potent matrices in Mn. Suppose ?:MnMn is a map satisfying A-λBΓn implies ?(A)-λ?(B)∈Γn, where A, BMn, λC. Then either ? is of the form ?(A)=cTAT-1, AMn, or ? is of the form ?(A)=cTAtT-1, AMn, where TMn is an invertible matrix, cC satisfies ck=c.  相似文献   

15.
In this paper we study the class of square matrices A such that AA − AA is nonsingular, where A stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA − AA is nonsingular if and only if R(A)R(A)=Cn,1, where R(·) denotes the range space. Also we study matrices A such that R(A)=R(A).  相似文献   

16.
We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Qij = 1 for 0 ? j − i ? N − 1 and Qij = 0 otherwise. We prove that det(QQ) is the minimum of det(RR) over all complex matrices R with the same dimensions as Q satisfying ∣Rij∣ ? 1 whenever Qij = 1 and Rij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ.  相似文献   

17.
Let A be an n×n matrix with eigenvalues λ1,λ2,…,λn, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA2 is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA2,‖A‖,‖AA-AA‖,n and m.  相似文献   

18.
The spectra of some trees and bounds for the largest eigenvalue of any tree   总被引:2,自引:0,他引:2  
Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let nkj+1 and dkj+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (nk = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d1d2, …, dk−1dk ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.  相似文献   

19.
Let Λn:={λ0<λ1<?<λn} be a set of real numbers. The collection of all linear combinations of eλ0t,eλ1t,…,eλnt over R will be denoted by
E(Λn):=span{eλ0t,eλ1t,…,eλnt}.  相似文献   

20.
Let M denote a 2 × 2 block complex matrix , where A and D are square matrices, not necessarily with the same orders. In this paper explicit representations for the Drazin inverse of M are presented under the condition that BDiC = 0 for i = 0, 1, … , n − 1, where n is the order of D.  相似文献   

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