On the dimension of the algebra generated by two positive semi-commuting matrices

Kandić, Marko; Šivic, Klemen

doi:10.1016/j.laa.2016.09.027

Cited by 4 publications

(11 citation statements)

References 19 publications

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“…4.7). Moreover, the paper [6] provides a nontrivial example showing that this upper bound can be attained. In our paper this result is proved in more transparent way that also gives some insight in constructing the just-mentioned example.…”

Section: Introductionmentioning

confidence: 97%

“…Recently, Kandić andŠivic [6] have studied order analogs of Gerstenhaber's theorem stating that the dimension of the unital algebra generated by two commuting n × n matrices is at most n. They showed that the dimension of the unital algebra generated by two positive n × n matrices A and B is at most n(n + 1)/2 provided its commutator [A, B] = AB − BA is also positive (see Theorem 3.1). Here positivity of a matrix means that it has nonnegative entries.…”

Section: Introductionmentioning

confidence: 99%

“…It has been also shown in [6] that 9 is the largest dimension of the unital algebra generated by two positive idempotent matrices E and F satisfying EF ≥ F E (see Corollary Date: November 13, 2018. 4.7).…”

Section: Introductionmentioning

confidence: 99%

“…More about super commutants can be found in [1].3. Positive matricesSearching for order analogs of Gerstenhaber's theorem, Kandić andŠivic[6] have recently proved the following theorem[6, Theorem 3.2]. In fact, this theorem also follows from[7, Proposition 4.3].…”

mentioning

confidence: 98%

“…This shows that the commutator of A and E has the formEA − AE = − EA) 2 = 0 = (AE − EA) 2 E.The main result of this section now easily follows. With a different proof it was already shown in[6, Theorem 6.6] under slightly weaker assumption, as the order continuity of the idempotent E was not needed.…”

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confidence: 99%

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On algebras generated by positive operators

Drnovšek

2017

Positivity

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Abstract. We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandić andŠivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices E and F such that EF ≥ F E is equal to the linear span of the set {I, E, F, EF, F E, EF E, F EF, (EF ) 2 , (F E) 2 }, and so its dimension is at most 9. We give examples of two positive idempotent matrices that generate unital algebra of dimension 2n if n is even, and of dimension (2n − 1) if n is odd.We also prove that the algebra generated by positive matrices B 1 , B 2 , . . ., B k is triangularizable if AB i ≥ B i A (i = 1, 2, . . . , k) for some positive matrix A with distinct eigenvalues.

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Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

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On algebras generated by positive operators

Drnovšek

2017

Positivity

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On real algebras generated by positive and nonnegative matrices

Kolegov

2021

Linear Algebra and its Applications

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A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

Kurniadi

2023

BAREKENG: J. Math. & App.

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There are three conditions for a topological space to be said a topological manifold of dimension : Hausdorff space, second-countable, and the existence of homeomorphism of a neighborhood of each point to an open subset of or -dimensional locally Euclidean. The differentiable structure is given if the intersection of two charts is an empty chart or its transition map is differentiable. In this article, we study a differentiable manifold on finite dimensional real vector spaces. The aim is to prove that any finite-dimensional vector space is a differentiable manifold. First of all, it is proved that a finite dimensional vector space is a topological manifold by constructing a norm as its topology. Given a metric which is induced by a norm. Two norms on a finite dimensional vector space are always equivalent and they are determine the same topology. Secondly, it is proved that the transition map in the finite dimensional vector space is differentiable. As conclusion, we have that any finite dimensional vector space with independent norm topology choice is a differentiable manifold. As a matter of discussion, it can be studied that the vector space of all linear operators of a finite dimensional vector space has a differentiable manifold structure as well.

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On the dimension of the algebra generated by two positive semi-commuting matrices

Cited by 4 publications

References 19 publications

On algebras generated by positive operators

On algebras generated by positive operators

On real algebras generated by positive and nonnegative matrices

A Differentiable Structure on a Finite Dimensional Real Vector Space as a Manifold

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