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1.
We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A1,A2, an n × n nilpotent Toeplitz matrix Nn, and an n × n cyclic permutation matrix Sn(s) such that the numbers of flat portions on the boundaries of W(A1⊕Nn) and W(A2⊕Sn(s)) are, respectively, 2(n - 2) and 2n. 相似文献
2.
Thomas H. Pate 《Linear and Multilinear Algebra》2003,51(3):263-278
If 1≤k≤n, then Cor(n,k) denotes the set of all n×n real correlation matrices of rank not exceeding k. Grone and Pierce have shown that if A∈Cor (n, n-1), then per(A)≥n/(n-1). We show that if A∈Cor(n,2), then , and that this inequality is the best possible. 相似文献
3.
Let A be an n × n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A. 相似文献
4.
Let Knbe the convex set of n×npositive semidefinite doubly stochastic matrices. If Aε kn, the graph of A,G(A), is the graph on n vertices with (i,j) an edge if aij ≠ 0i≠ j. We are concerned with the extreme points in Kn. In many cases, the rank of Aand G(A) are enough to determine whether A is extreme in Kn. This is true, in particular, if G(A)is a special kind of nonchordal graph, i.e., if no two cycles in G(A)have a common edge. 相似文献
5.
Let A be a matrix in
r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {πn(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that πn(A,1/2) (x) = O(n(A)/2lnr−1(n)) and πn+1(A,1/2) (x) − πn(A,1/2) (x) = O(n((A)−1)/2lnr−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}. 相似文献
6.
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
7.
Roy Meshulam 《Linear and Multilinear Algebra》1990,26(1):39-41
It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1. 相似文献
8.
M. Barraa 《Linear algebra and its applications》2002,350(1-3):289-292
Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δS,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δS,A,B is convexoid if and only if A and B are convexoid. 相似文献
9.
Si-Ju Kim 《Linear and Multilinear Algebra》1997,42(4):319-322
A square matrix A with per A≠0 is called convertible if there exists a (1, -1) matrix H such that per A=det(H o A) where H o A denote the Hadamard product of H and A In this paper, an upper bound of permanents of maximal convertible (0,1) matrices A with π(A)≥4(n-1) is obtained. 相似文献
10.
Let us denote ab=max(a,b) and ab=a+b for
and extend this pair of operations to matrices and vectors in the same way as in linear algebra. We present an O(n2(m+n log n)) algorithm for finding all essential terms of the max-algebraic characteristic polynomial of an n×n matrix over
with m finite elements. In the cases when all terms are essential, this algorithm also solves the following problem: Given an n×n matrix A and k{1,…,n}, find a k×k principal submatrix of A whose assignment problem value is maximum. 相似文献
11.
A function, F, on the space of n×n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument, that is F(A)=F(UAUT) for every orthogonal U and symmetric A in its domain. Spectral functions are in one-to-one correspondence with the symmetric functions on
: those that are invariant under arbitrary swapping of their arguments. In this paper we show that a spectral function has a quadratic expansion around a point A if and only if its corresponding symmetric function has quadratic expansion around λ(A) (the vector of eigenvalues). We also give a concise and easy to use formula for the ‘Hessian' of the spectral function. In the case of convex functions we show that a positive definite ‘Hessian' of f implies positive definiteness of the ‘Hessian' of F. 相似文献
12.
Morris Newman 《Linear and Multilinear Algebra》1982,11(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
13.
M. H. Lim 《Linear and Multilinear Algebra》1979,7(2):145-147
In this note we prove that there is no linear mapping T on the space of n-square symmetric matrices over any subfield of real field such that the determinant of A is equal to the permanent of T(A) for all symmetric matrices A if n≥3. 相似文献
14.
Suppose A∈Mn×m(F), B∈Mn×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB). 相似文献
15.
Let A be a 0, 1-matrix with at most one 1 in each row and column. The authors prove that the numerical range of A is the convex hull of a polygon and a circular disk, both centered at the origin and contained in the unit disk. The proof uses a permutation similarity to reduce A to a direct sum of matrices whose numerical ranges can be determined. A computer program, developed by the authors, which plots the boundary of the numerical range of an arbitrary complex matrix is also discussed. 相似文献
16.
Inertially arbitrary patterns 总被引:11,自引:0,他引:11
An n×n sign pattern matrix A is an inertially arbitrary pattern (IAP) if each non-negative triple (rst) with r+s+t=n is the inertia of a matrix with sign pattern A. This paper considers the n×n(n≥2) skew-symmetric sign pattern Sn with each upper off-diagonal entry positive, the (1,1) entry negative, the (nn) entry positive, and every other diagonal entry zero. We prove that Sn is an IAP. 相似文献
17.
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
18.
A matrix A in the semigroup Nn of non-negative n×nmatrices is prime if A is not monomial and A=BC,BCεNn implies that either B or C is monomial. One necessary and another sufficient condition are given for a matrix in Nn to be prime. It is proved that every prime in Nn is completely decomposable. 相似文献
19.
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
20.
We study here some linear recurrence relations in the algebra of square matrices. With the aid of the Cayley–Hamilton Theorem, we derive some explicit formulas for An (nr) and etA for every r×r matrix A, in terms of the coefficients of its characteristic polynomial and matrices Aj, where 0jr−1. 相似文献