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1.
In this paper we classify linear maps preserving commutativity in both directions on the space N(F) of strictly upper triangular (n+1)×(n+1) matrices over a field F. We show that for n3 a linear map on N(F) preserves commutativity in both directions if and only if =′+f where ′ is a product of standard maps on N(F) and f is a linear map of N(F) into its center. 相似文献
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In this note, we show that the set of all commuting d-tuples of commuting n×n matrices that are contained in an n-dimensional commutative algebra is a closed set, and therefore, Gerstenhaber's theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs. We show that the variety of commuting triples of 4×4 matrices is irreducible. We also study the variety of n-dimensional commutative subalgebras of Mn(F), and show that it is irreducible of dimension n2−n for n4, but reducible, of dimension greater than n2−n for n7. 相似文献
4.
C. W. Norman 《Linear and Multilinear Algebra》1995,38(4):351-371
Let A, B denote the companion matrices of the polynomials xm,xn over a field F of prime order p and let λ,μ be non-zero elements of an extension field K of F. The Jordan form of the tensor product (λI + A)⊗(μI + B) of invertible Jordan matrices over K is determined via an equivalent study of the nilpotent tranformation S of m × n matrices X over F where(X)S = A TX + XB. Using module-theoretic concepts a Jordan basis for S is specified recursively in terms of the representations of m and n in the scale of p, and reduction formulae for the elementary divisors of S are established. 相似文献
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Let T be a linear operator on the space of all m×n matrices over any field. we prove that if T maps rank-2 matrices to rank-2 matrices then there exist nonsingular matrices U and V such that either T(X)=UXV for all matrices X, or m=n and T(X)=UXtV for all matrices X where Xt denotes the transpose of X. 相似文献
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Raphael Loewy 《Linear and Multilinear Algebra》1992,33(1):23-30
Let Vdenote either the space of n×n hermitian matrices or the space of n×nreal symmetric matrices, Given nonnegative integers r,s,t such that r+S+t=n, let G( r,s,r) denote the set of all matrices in V with inertia (r,s,t). We consider here linear operators on V which map G(r,s,t) into itself. 相似文献
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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
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Joaquim F. Machado 《Linear and Multilinear Algebra》1995,40(1):33-35
Let F be an algebraically closed field. We denote by i(A) the number of invariant polynomials of a square matrix A, which are different from 1. For A,B any n×n matrices over F, we calculate the maximum of i(XAX-1+B), where X runs over the set of all non-singular n×n matrices over F. 相似文献
9.
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
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Stephen Piercet 《Linear and Multilinear Algebra》1979,8(2):101-114
Consider the n-square matrices over an infiniie field Kas an n2-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK) 相似文献