共查询到20条相似文献,搜索用时 31 毫秒
1.
Kent Griffin Jeffrey L. Stuart Michael J. Tsatsomeros 《Czechoslovak Mathematical Journal》2008,58(4):1185-1193
Let a, b and c be fixed complex numbers. Let M
n
(a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ k ⩽ n, each k × k principal minor of M
n
(a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of
M
n
(a, b, c). We also show that all complex polynomials in M
n
(a, b, c) are Toeplitz matrices. In particular, the inverse of M
n
(a, b, c) is a Toeplitz matrix when it exists. 相似文献
2.
Wei Fang Wei Gao Yubin Gao Fei Gong Guangming Jing Zhongshan Li Yanling Shao Lihua Zhang 《Czechoslovak Mathematical Journal》2016,66(3):895-911
A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set {+,?, 0} ({+, 0}, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix A is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of A. Using a correspondence between sign patterns with minimum rank r ≥ 2 and point-hyperplane configurations in Rr?1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established. 相似文献
3.
In Hyoun Kim 《Journal of Mathematical Analysis and Applications》2007,335(1):260-267
In this paper it is shown that the spectrum σ, a set-valued function, is continuous when the function is restricted to the set of all ‘quasi-n-hyponormal’ operators acting on an infinite-dimensional separable Hilbert space, where a quasi-n-hyponormal operator is defined to be unitarily equivalent to an n×n upper triangular operator matrix whose diagonal entries are hyponormal operators. 相似文献
4.
Kinkar Ch. Das 《Discrete Mathematics》2011,(22):2593
Let G=(V,E) be a simple, connected and undirected graph with vertex set V(G) and edge set E(G). Also let D(G) be the distance matrix of a graph G (Jane?i? et al., 2007) [13]. Here we obtain Nordhaus–Gaddum-type result for the spectral radius of distance matrix of a graph.A sharp upper bound on the maximal entry in the principal eigenvector of an adjacency matrix and signless Laplacian matrix of a simple, connected and undirected graph are investigated in Das (2009) [4] and Papendieck and Recht (2000) [15]. Generally, an upper bound on the maximal entry in the principal eigenvector of a symmetric nonnegative matrix with zero diagonal entries and without zero diagonal entries are investigated in Zhao and Hong (2002) [21] and Das (2009) [4], respectively. In this paper, we obtain an upper bound on minimal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. Moreover, we present the lower and upper bounds on maximal entry in the principal eigenvector for the distance matrix of a graph and characterize extremal graphs. 相似文献
5.
Kh. D. Ikramov 《Mathematical Notes》1996,60(6):649-657
LetR be a (real or complex) triangular matrix of ordern, say, an upper triangular matrix. Is it true that there exists a normaln×n matrixA whose upper triangle coincides with the upper triangle ofR? The answer to this question is “yes” and is obvious in the following cases: (1)R is real; (2)R is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries ofR belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrixR of order 2. However, even forn=3 this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for 3×3 matrices. 相似文献
6.
Frank Harary 《Numerische Mathematik》1962,4(1):128-135
LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P
–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D
* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD
* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA
* be the adjacency matrix ofD
*. It is easy to show that there exists a permutation matrixQ such thatQA
*
Q
–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961 相似文献
7.
Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered. 相似文献
8.
Bruce A. Barnes 《Mathematische Nachrichten》2002,241(1):5-20
Assume that T is an upper triangular square matrix with entries in a unital Banach algebra. The main question studied here is: Under what conditions on the entries in T is it true that the spectrum of T is the union of the spectra of the diagonal entries of T? Also some results are proved concerning the Fredholm theroy of matrices with operator entries. 相似文献
9.
Henryk Minc 《Linear and Multilinear Algebra》2013,61(1-2):91-94
It is shown that if all subpermaneats of order k of an n × n doubly stochastic matrix are equal for some k ≤ n ? 2, then all the entries of the matrix must be equal to 1/n. 相似文献
10.
John Brian Burghduff 《Linear and Multilinear Algebra》1995,40(2):125-140
The permanent function on the set of n×n doubly stochastic matrices with zero main diagonal attains a strict local minimum at the matrix whose off diagonal entries are all equal to 1/(n-1). 相似文献
11.
This paper is concerned with the interdependence of the irreducible constituents of an algebra of n × n matrices over a field F. It is shown that there is a similarity transformation reducing the algebra to a block triangular form in which, ateach pair of diagonal places, the blocks either are always equal or may be occupied by any entries from the corresponding irreducible constituents. A recent theorem of Kaplan-sky is extended as an application of this result. 相似文献
12.
Kelly Wieand 《Journal of Theoretical Probability》2003,16(3):599-623
We consider a class of random matrix ensembles which can be constructed from the random permutation matrices by replacing the nonzero entries of the n×n permutation matrix matrix with M×M diagonal matrices whose entries are random Kth roots of unity or random points on the unit circle. Let X be the number of eigenvalues lying in a specified arc I of the unit circle, and consider the standardized random variable (X–E[X])/(Var(X))1/2. We show that for a fixed set of arcs I
1,...,I
N
, the corresponding standardized random variables are jointly normal in the large n limit, and compare the covariance structures which arise with results for other random matrix ensembles. 相似文献
13.
A lower triangular matrix with nonzero principal diagonal entries is called a triangle. In this paper we obtain the sufficient conditions for ∑anλn to be summable ∣A∣k whenever ∑an is summable ∣T∣k for a triangle T. 相似文献
14.
Forumulas are given for all of the eigenvalues and eigenvectors of the distance matrix of the path Pn on n vertices. It is shown that Pn has the maximum distance spectral radius among all connected graphs of order n, and an ordering property of the entries of the Perron-Frobenius eigenvector is presented. 相似文献
15.
Lowell Abrams Donniell E. Fishking Silvia Valdes-Leon 《Linear and Multilinear Algebra》2013,61(2):129-136
Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let BR denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B-1 ≡ BR mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices. 相似文献
16.
Shmuel Friedland 《Linear algebra and its applications》1975,12(2):127-137
Let A be an n×n complex-valued matrix, all of whose principal minors are distinct from zero. Then there exists a complex diagonal matrix D, such that the spectrum of AD is a given set σ = {λ1,…,λn} in C. The number of different matrices D is at most n!. 相似文献
17.
A real symmetric matrix of order n, n ? 2, is said to be paramount if each proper principal minor is not less than the absolute value of any other minor built from the same rows. A paramount matrix is minimal 1 if reducing any of the diagonal entries removes the matrix from the paramount class. Minimal paramount matrices arise in the n-port realization problem of circuit theory. A condition is found that is equivalent to the minimality of a paramount matrix. Conditions are also found that guarantee that the inverse of an invertible minimal paramount matrix is itself minimal. 相似文献
18.
We show that a semigroup of positive matrices (all entries greater than or equal to zero) with binary diagonals (diagonal
entries either 0 or 1) is either decomposable (all matrices in the semigroup have a common zero entry) or is similar, via
a positive diagonal matrix, to a binary semigroup (all entries 0 or 1). In the case where the idempotents of minimal rank
in S{\mathcal{S}} satisfy a “diagonal disjointness” condition, we obtain additional structural information. In the case where the semigroup
is not necessarily positive but has binary diagonals we show that either the semigroup is reducible or the minimal rank ideal
is a binary semigroup. We also give generalizations of these results to operators acting on the Hilbert space of square-summable
sequences. 相似文献
19.
We investigate classes of real square matrices possessing some weakened from of strict diagonal dominance of a real matrix whose diagonal entries are all positive. The intersection of each one of these classes with the set of all real matrices, with nonpositive off-diagonal elements, coincides with the set of all nonsingular M- matrices. 相似文献
20.
The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined. 相似文献