Coherence invariant maps on tensor products

Chooi, Wai Leong; Kwa, Kiam Heong; Lim, Ming-Huat

doi:10.1016/j.laa.2016.11.023

Cited by 8 publications

(3 citation statements)

References 9 publications

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“…Hua's theorem also characterized the graph isomorphisms on rectangular matrices, and his work was continued by many scholars (cf. [3], [4], [11]- [17], [20]- [27]). Thus, the basic problem of the geometry of matrices is also to study graph isomorphisms and graph homomorphisms on matrices.…”

Section: Clearly Ad(a B) ≥ 0 Ad(a B) = 0 ⇔ a = B; Ad(a B) = Ad(b A); ...mentioning

confidence: 99%

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Graph homomorphisms on rectangular matrices over division rings II

Huang

Zhao

2018

Linear and Multilinear Algebra

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Let D m×n be the set of m × n matrices over a division ring D. Two matrices A, B ∈ D m×n are adjacent if rank(A − B) = 1. By the adjacency, D m×n is a connected graph. Suppose D, D ′ are division rings and m, n, m ′ , n ′ ≥ 2 are integers. We determine additive graph homomorphisms from D m×n to D ′m ′ ×n ′ . When |D| ≥ 4, we characterize the graph homomorphism ϕ : D n×n → D ′m ′ ×n ′ if ϕ(0) = 0 and there exists A 0 ∈ D n×n such that rank(ϕ(A 0 )) = n. We also discuss properties and ranges on degenerate graph homomorphisms. If f : D m×n → D ′m ′ ×n ′ (where min{m, n} = 2) is a degenerate graph homomorphism, we prove that the image of f is contained in a union of two maximal adjacent sets of different types. For the case of finite fields, we obtain two better results on degenerate graph homomorphisms.Using Lemmas 2.1 and 1.1, we can prove the following results. Corollary 2.2 (cf. [26, Corollary 3.10]) Let A and B be two adjacent points in D m×n . Then there are exactly two maximal sets M and M ′ containing A and B. Moreover, M and M ′ are of different types. Corollary 2.3 (cf. [9, 12, 26]) If M and N are two distinct maximal sets of the same type [resp. different types] in D m×n with M ∩ N ∅, then |M ∩ N| = 1 [resp. |M ∩ N| ≥ 2].Lemma 2.4 (cf. [14, Lemma 3.4]) Suppose that M, N are two distinct maximal sets in D m×n with M ∩ N ∅. Then: (i) if M, N are of different types, then for any A ∈ M ∩ N, there are P ∈ GL m (D) and Q ∈ GL n (D) such that M = PM 1 Q + A = PM 1 + A and N = PN 1 Q + A = N 1 Q + A;

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Section: Clearly Ad(a B) ≥ 0 Ad(a B) = 0 ⇔ a = B; Ad(a B) = Ad(b A); ...mentioning

confidence: 99%

“…Hua's theorem also characterized the graph isomorphism on rectangular matrices, and his work was continued by many scholars (cf. [3], [9]- [19], [21], [23]- [28]).…”

Section: Introductionmentioning

confidence: 99%

Graph homomorphisms on rectangular matrices over division rings II

Huang

Zhao

2018

Linear and Multilinear Algebra

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“…Denote by π k the projection of φ on direction k (k ∈ [[m]]). Then by Corollary 3.10 of[3] we haveφ(A) = R j=1 φ(α 1j × α 2j × . .…”

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confidence: 97%

Commutation matrices and commutation tensors

Lin

2018

Linear and Multilinear Algebra

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The commutation matrix was first introduced in statistics as a transposition matrix by Murnaghan in 1938. In this paper, we first investigate the commutation matrix which is employed to transform a matrix into its transpose. We then extend the concept of the commutation matrix to commutation tensor and use the commutation tensor to achieve the unification of the two formulae of the linear preserver of the matrix rank, a classical result of Marcus in 1971.

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Adjacency preserving maps between exterior powers of vector spaces

Lau,

Lim

2024

Linear Algebra and its Applications

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Coherence invariant maps on tensor products

Cited by 8 publications

References 9 publications

Graph homomorphisms on rectangular matrices over division rings II

Graph homomorphisms on rectangular matrices over division rings II

Commutation matrices and commutation tensors

Adjacency preserving maps between exterior powers of vector spaces

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