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1.
Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC-1for some C = C-1over F) in the following cases: (1)B= aI - Afor some a ε F and (2) B = A-1. Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.  相似文献   

2.
Let A be an integral matrix such that det A = 1 mod mAAT mod m, where m is odd. It is shown that a symmetric integral matrix B of determinant 1 exists such that BA mod m. The result is false if m is even.  相似文献   

3.
In this article we provide generalizations of Specht's theorem which states that two n × n matrices A and B are unitarily equivalent if and only if all traces of words in two non-commuting variables applied to the pairs (A, A*) and (B, B*) coincide. First, we obtain conditions which allow us to extend this to simultaneous similarity or unitary equivalence of families of operators, and secondly, we show that it suffices to consider a more restricted family of functions when comparing traces. Our results do not require the traces of words in (A, A*) and (B, B*) to coincide, but only to be close.  相似文献   

4.
The principal results are that if A is an integral matrix such that AAT is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.  相似文献   

5.
A theorem of the alternatives for the equation Ax + B|x| = b   总被引:4,自引:0,他引:4  
The following theorem is proved: given square matrices A, D of the same size, D nonnegative, then either the equation Ax + B|x| = b has a unique solution for each B with |B| ≤ D and for each b, or the equation Ax + B0|x| = 0 has a nontrivial solution for some matrix B0 of a very special form, |B0| ≤ D; the two alternatives exclude each other. Some consequences of this result are drawn. In particular, we define a λ to be an absolute eigenvalue of A if |Ax| = λ|x| for some x ≠ 0, and we prove that each square real matrix has an absolute eigenvalue.  相似文献   

6.
A derivation for the kernel of the irreducible representation T(λ) of the general linear group GLn(C) is given. This is then applied to the problem of determining necessary and sufficient conditions under which T(λ)(A) = T(λ)(B), where A and B are linear transformations, not necessarily invertible. Finally, conditions are obtained under which normality of T(λ)(A) implies normality of A.  相似文献   

7.
《Discrete Mathematics》1982,40(2-3):277-284
This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:BnB can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBFn[A]. In this paper the following questions are presented: (1) What is the relationship between GBFn[A1] and GBFn[A2] when A1A2. (2) What can be said about GBFn[A1A2] and GBFn[A1A2] in comparison with GBFn[A1]∩GBFn[A2] and GBFn[A1]GBFn[A2], respectively.  相似文献   

8.
Let Rbe a principal ideal ringRn the ring of n× nmatrices over R, and dk(A) the kth determinantal divisor of Afor 1 ≤ kn, 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   

9.
Given a pair of n×n matricesA and B, one may form a polynomial P(A,B,λ) which generalizes the characteristic polynomial of BP(B,λ). In particular, when A=I (identity), P(A, B,λ) = P(B,λ), the characteristic polynomial of B. C. Johnson has conjectured [1] (among other things) that when A and B are hermitian and A is positive definite, then P(A,B,λ) has real roots. The case n=2 can be done by hand. In this paper we verify the conjecture for n=3.  相似文献   

10.
It is shown that if A[ω] is a principal submatrix of the positive definite Hermitian matrix A, then A -1[ω] -(A[ω])-1is a positive semidefinite hermitian matrix. This fact is used to give a brief proof of a result of Saburou Saitoh concerning Hadamard products.  相似文献   

11.
Given a pair of matrices (AB) it is well known that its invariant factors and its controllability indices form a complete set of invariants for the Γ-equivalence [11] or block similarity [5]. How do they vary by adding columns to B? This problem was solved in [12] when B = 0; here we give a complete answer for this question.  相似文献   

12.
Let a positive definite Hermitian matrix HεMn(C) be decomposed as H=A + iB, with A, B ε Mnm(R). We give two new proofs of the inequality det H ≤ det A (with equality iff B = 0. each of which vields something futher. One exhibits majorization between the eigenvalues of A and H the other allows proof of the permanental analog per H≥per A.  相似文献   

13.
In this note, we characterize those pairs of nonzero r-by-d complex matrices that satisfy N2(AB) = N2(A)N2(B), in which N2(·) is the spectral norm and · is the Hadamard product.  相似文献   

14.
Inequalities for M and N0-matrices   总被引:1,自引:0,他引:1  
In this paper, we study inequalities for two M or N0-matrices A,B with AB. Necessary and sufficient conditions are given for positive linear combination of A and B to be an N0-matrix.  相似文献   

15.
Let R be a principal ideal ringRn the ring of n × n it matrices over R. It is shown that if A, B, X, Y are elements of R* such that A = XB, B = YA, then A and B are left equivalent. Some consequences are given.  相似文献   

16.
The Sylvester “inivilator” equation AX - XB=C is solved in finite terms when A and B are quite general. Newman's formula for the case of A and B periodic is recovered by a limiting process. Some applications are discussed.  相似文献   

17.
Let A be a matrixp(x) a polynomial. Put B=p(A). It is shown that necessary and sufficient conditions for A to be a polynomial in B are (i) if λ is any eigenvalue of A, and if some elementary divisor of A corresponding to λ is nonlinear, thenp'(λ)≠0;and (ii) if λ,μ are distinct eigenvalues of A, then p(λ)p(μ) are also distinct. Here all computations are over some algebraically closed field.  相似文献   

18.
Many different fuzzy implication operators have been proposed; most of them fit into one of the two classes: implication operations that are based on an explicit representation of implication AB in terms of &, , and ¬ (e.g., S-implications that are based on the formula B ¬ A), and R-implications that are based on an implicit representation of implication AB as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as ba) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, , and ¬ is described by (implicit) axioms.  相似文献   

19.
If AB are n × n M matrices with dominant principal diagonal, we show that 3[det(A + B)]1/n ≥ (det A)1/n + (det B)1/n.  相似文献   

20.
Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ :A →A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B∈ A, where[A,B]3 is the 3-commutator of A,B defined by[A, B]3 = [[[A, B],B],B] with [A,B] = AB-BA. The main result in this paper is shown that.,if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h : A →F and a scalar λ∈ F with λ~4 = 1 such that Φ(A)=λ A+h(A)I for all A ∈ A.  相似文献   

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