共查询到20条相似文献,搜索用时 171 毫秒
1.
Fernando C. Silva 《Linear and Multilinear Algebra》2013,61(3-4):269-277
Let A and B be n × n nonsingular matrices over a field F, and c 1,…,c n ∈ F. We give a necessary and sufficient condition for the existence of matrices A′ and B′ similar to A and B, respectively, such that A′ B′ has eigenvalues c 1,…,c n. 相似文献
2.
F. Conceição Silva 《Linear and Multilinear Algebra》2013,61(1):7-13
Let A and B be n×n matrices over a field F, and c 1,…,cn ∈F. We give a sufficient condition for the existence of matrices A' and B' similar to A and B, respectively, such that A' + B' has eigenvalues c 1,…,cn . 相似文献
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Helena ?migoc 《Linear and Multilinear Algebra》2013,61(2):85-96
This article presents a technique for combining two matrices, an n?×?n matrix M and an m?×?m matrix B, with known spectra to create an (n?+?m???p)?×?(n?+?m???p) matrix N whose spectrum consists of the spectrum of the matrix M and m???p eigenvalues of the matrix B. Conditions are given when the matrix N obtained in this construction is nonnegative. Finally, these observations are used to obtain several results on how to construct a realizable list of n?+?1 complex numbers (λ1,λ2,λ3,σ) from a given realizable list of n complex numbers (c 1,c 2,σ), where c 1 is the Perron eigenvalue, c 2 is a real number and σ is a list of n???2 complex numbers. 相似文献
5.
Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d. 相似文献
6.
According to a long standing conjecture, the geometric location of eigenvalues of doubly stochastic matrices of order n is exactly the union of regular k-gons anchored at 1 in the unit disc for 2 ≤ k ≤ n. It is easy to verify this fact for n?=?2,?3. But, for n?≥?4, it has been an open question. We show that this conjecture is wrong for n?=?5. 相似文献
7.
Lin Chen 《Linear and Multilinear Algebra》2013,61(6):725-730
Let M n (𝔸) and T n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T n (𝔸) into M n (𝔸) is of the form A?→?AT???TA?+?A ??+?ξ(A)I n , where T?∈?M n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T n (𝔸) and A ? is the image of A under???applied entrywise. 相似文献
8.
Abraham Berman Christopher King Robert Shorten 《Linear and Multilinear Algebra》2013,61(10):1117-1123
In this article, a general notion of common diagonal Lyapunov matrix is formulated for a collection of n?×?n matrices A 1,?…?,?A s , and cones k 1,?…?,?k s in ? n . Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting. The conditions are similar to and extend the well-known criteria for the case s?=?1, k 1?=?? n . 相似文献
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Susana Furtado 《Linear algebra and its applications》2008,429(7):1663-1678
Let F be a field. In [Djokovic, Product of two involutions, Arch. Math. 18 (1967) 582-584] it was proved that a matrix A∈Fn×n can be written as A=BC, for some involutions B,C∈Fn×n, if and only if A is similar to A-1. In this paper we describe the possible eigenvalues of the matrices B and C.As a consequence, in case charF≠2, we describe the possible similarity classes of (P11⊕P22)P-1, when the nonsingular matrix P=[Pij]∈Fn×n, i,j∈{1,2} and P11∈Fs×s, varies.When F is an algebraically closed field and charF≠2, we also describe the possible similarity classes of [Aij]∈Fn×n, i,j∈{1,2}, when A11 and A22 are square zero matrices and A12 and A21 vary. 相似文献
11.
Relative perturbation bounds for the unitary polar factor 总被引:5,自引:0,他引:5
Ren-Cang Li 《BIT Numerical Mathematics》1997,37(1):67-75
LetB be anm×n (m≥n) complex (or real) matrix. It is known that there is a uniquepolar decomposition B=QH, whereQ*Q=I, then×n identity matrix, andH is positive definite, providedB has full column rank. Existing perturbation bounds suggest that in the worst case, for complex matrices the change inQ be proportional to the reciprocal ofB's least singular value, or the reciprocal of the sum ofB's least and second least singular values if matrices are real. However, there are situations where this unitary polar factor is much more accurately determined by the data than the existing perturbation bounds would indicate. In this paper the following question is addressed: how much mayQ change ifB is perturbed to $\tilde B = D_1^* BD_2 $ , whereD 1 andD 2 are nonsingular and close to the identity matrices of suitable dimensions? It is shown that for a such kind of perturbation, the change inQ is bounded only by the distances fromD 1 andD 2 to identity matrices and thus is independent ofB's singular values. Such perturbation is restrictive, but not unrealistic. We show how a frequently used scaling technique yields such a perturbation and thus scaling may result in better-conditioned polar decompositions. 相似文献
12.
Morris Newman 《Linear and Multilinear Algebra》2013,61(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
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Li Qiu 《Linear algebra and its applications》2007,422(1):304-307
Let A1, … , Ak be positive semidefinite matrices and B1, … , Bk arbitrary complex matrices of order n. We show that
span{(A1x)°(A2x)°?°(Akx)|x∈Cn}=range(A1°A2°?°Ak) 相似文献
16.
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.
17.
Let X ? denotes the Moore--Penrose pseudoinverse of a matrix X. We study a number of situations when (aA?+?bB)??=?aA?+?bB provided a,?b?∈?????{0} and A, B are n?×?n complex matrices such that A ??=?A and B ??=?B. 相似文献
18.
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
19.
Using techniques from algebraic topology we derive linear inequalities which relate the spectrum of a set of Hermitian matrices A1,…, Ar ? ¢n×n with the spectrum of the sum A1 + … + Ar. These extend eigenvalue inequalities due to Freede-Thompson and Horn for sums of eigenvalues of two Hermitian matrices. 相似文献
20.
Raphael Loewy 《Linear and Multilinear Algebra》2013,61(4):355-382
Let k and n be positive integers such that kn. 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 Ak for every A?L. 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 x?W A?L. We show here, under some mild assumptions on k n 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