On determinant preserver problems

Tan, Victor; Wang, Fei

doi:10.1016/s0024-3795(02)00739-5

Cited by 31 publications

(19 citation statements)

References 3 publications

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“…Linear maps, additive maps, and multiplicative maps of determinants and functions of eigenvalues have been studied by researchers; see [6,7,16]. In this section, we study Schur multiplicative maps preserving determinant and related functions.…”

Section: Preservers Of Determinant and Other Functions Of Eigenvaluesmentioning

confidence: 99%

Schur Multiplicative Maps on Matrices

Clark

Rastogi

2008

Bulletin of the Australian Mathematical Society

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The structure of Schur multiplicative maps on matrices over a field is studied. The result is then used to characterize Schur multiplicative maps f satisfying f (S) ⊆ S for different subsets S of matrices including the set of rank k matrices, the set of singular matrices, and the set of invertible matrices. Characterizations are also obtained for maps on matrices such that ( f (A)) = (A) for various functions including the rank function, the determinant function, and the elementary symmetric functions of the eigenvalues. These results include analogs of the theorems of Frobenius and Dieudonné on linear maps preserving the determinant functions and linear maps preserving the set of singular matrices, respectively.2000 Mathematics subject classification: 15A03, 15A15.

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Section: Preservers Of Determinant and Other Functions Of Eigenvaluesmentioning

confidence: 99%

Schur Multiplicative Maps on Matrices

Clark

Rastogi

2008

Bulletin of the Australian Mathematical Society

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“…Recently, Dolinar andŠemrl [2] and Tan and Wang [4] respectively modified the problem by removing the linearity of φ and changing condition (1.1) to det(φ(A) + λφ(B)) = det(A + λB), for all A, B ∈ M n (F) and all λ ∈ F. This paper is motivated by [2] and [4]. We study determinant preservers on the vector space of symmetric matrices.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Because the proofs of Theorems 1.2 and 1.3 are similar to those for Theorems 2 and 3 of [4], they are omitted. In section 2 we will prove Theorem 1.1.…”

Section: Chongguang Cao and Xiaomin Tangmentioning

confidence: 99%

Determinant preserving transformations on symmetric matrix spaces

Cao¹,

Tang²

2004

ELA

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Abstract. Let Sn(F) be the vector space of n × n symmetric matrices over a field F (with certain restrictions on cardinality and characteristic). The transformations φ on the space which satisfy one of the following conditions:1. det(A + λB) = det(φ(A) + λφ(B)) for all A, B ∈ Sn(F) and λ ∈ F; 2. φ is surjective and det(A + λB) = det(φ(A) + λφ(B)) for all A, B and two specific λ; 3. φ is additive and preserves determinant are characterized.

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“…It is proved in [11] that Theorem A holds true without the surjectivity condition. Theorem A concerns with the determinant preserving maps with respect to the vector space structure of the underlying algebra.…”

Section: Introductionmentioning

confidence: 99%

DETERMINANT AND SPECTRUM PRESERVING MAPS ON M_n

Kim¹

2012

Korean Journal of Mathematics

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Abstract. Let M n be the algebra of all complex n × n matrices and ϕ : M n → M n a surjective map (not necessarily additive or multiplicative) satisfying one of the following equations:Then it is an automorphism, where σ(A) is the spectrum of A ∈ M n . We also show that if A be a standard operator algebra, B is a unital Banach algebra with trivial center and if ϕ : A → B is a multiplicative surjection preserving spectrum, then ϕ is an algebra isomorphism.

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On determinant preserver problems

Cited by 31 publications

References 3 publications

Schur Multiplicative Maps on Matrices

Schur Multiplicative Maps on Matrices

Determinant preserving transformations on symmetric matrix spaces

DETERMINANT AND SPECTRUM PRESERVING MAPS ON M_n

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On determinant preserver problems

Cited by 31 publications

References 3 publications

Schur Multiplicative Maps on Matrices

Schur Multiplicative Maps on Matrices

Determinant preserving transformations on symmetric matrix spaces

DETERMINANT AND SPECTRUM PRESERVING MAPS ON Mn

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DETERMINANT AND SPECTRUM PRESERVING MAPS ON M_n