An elementary method to compute the algebra generated by some given matrices and its dimension

Pascoe, J. E.

doi:10.1016/j.laa.2019.02.021

Cited by 11 publications

(8 citation statements)

References 9 publications

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“…So, the theory of finite-dimensional algebras can be applied. Also we continue investigations of generators of matrix algebras, see works [10,15,20] for some recent results on this topic.…”

Section: Introductionmentioning

confidence: 90%

On real algebras generated by positive and nonnegative matrices

Kolegov

2020

Preprint

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Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of nonnegative matrices up to similarity. Also we find all realizable dimensions of algebras generated by two nonnegative semi-commuting matrices. The last result provides the solution to the problem posed by M. Kandić, K. Sivic (2017) [13].

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“…So, the theory of finite-dimensional algebras can be applied. Also we continue investigations of generators of matrix algebras, see works [10,15,20] for some recent results on this topic.…”

Section: Introductionmentioning

confidence: 90%

On real algebras generated by positive and nonnegative matrices

Kolegov

2020

Preprint

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“…We could just as easily use Equation (2.4) as the definition of the 𝜓-involution. The 𝜓-involution was introduced by the third named author in [Pas19], where its key properties (including the modularity property) were described; we have included proofs here for the sake of convenience. What we will call the elementary Pick matrix, defined in the next section, also appears in [Pas19].…”

Section: Preliminaries the 𝜓 Involution And Its Propertiesmentioning

confidence: 99%

“…, 𝑋 𝑑 . By the main Theorem of [Pas19], we know that a matrix 𝑍 is in the algebra generated by 𝑋 if and only if vec(𝑍) ∈ ran(𝑃 𝑋 ). When each block of 𝑌 belongs to this algebra, then there will exist an nc polynomial matrix 𝑓 with 𝑓 (𝑋) = 𝑌 .…”

Section: Noncommutative Pick Interpolation and The Matrix 𝑃mentioning

confidence: 99%

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Effective noncommutative Nevanlinna-Pick interpolation in the row ball, and applications

Augat¹,

Jury²,

Pascoe³

2020

Preprint

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We provide an effective single-matrix criterion, in terms of what we call the elementary Pick matrix, for the solvability of the noncommutative Nevanlinna-Pick interpolation problem in the row ball, and provide some applications. In particular we show that the so-called "column-row property" fails for the free semigroup algebras, in stark contrast to the analogous commutative case. Additional applications of the elementary Pick matrix include a local dilation theorem for matrix row contractions and interpolating sequences in the noncommutative setting. Finally we present some numerical results related to the failure of the column-row property.

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“…It is especially useful given the following proposition, which shows that the ψ involution has a rich algebraic structure. The ψ involution was used to develop algorithms for understanding finite dimensional matrix algebras [17].…”

Section: Preliminariesmentioning

confidence: 99%

The outer spectral radius and dynamics of completely positive maps

Pascoe¹

2019

Preprint

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We examine a special case of an approximation of the joint spectral radius given by Blondel and Nesterov, which we call the outer spectral radius. The outer spectral radius is given by the square root of the ordinary spectral radius of the n 2 by n 2 matrix X i ⊗ X i . We give an analogue of the spectral radius formula for the outer spectral radius which can be used to quickly obtain the error bounds in methods based on the work of Blondel and Nesterov. The outer spectral radius is used to analyze the iterates of a completely positive map, including the special case of quantum channels. The average of the iterates of a completely positive map approach to a completely positive map where the Kraus operators span an ideal in the algebra generated by the Kraus operators of the original completely positive map. We also give an elementary treatment of Popescu's theorems on similarity to row contractions in the matrix case, describe connections to the Parrilo-Jadbabaie relaxation, and give a detailed analysis of the maximal spectrum of a completely positive map. Contents J. E. PASCOE 5. The outer spectrum and dynamics of completely positive maps 16 5.1. Maximum eigenvalue is nonnegative 17 5.2. Structure of the T λ 18 5.3. Spectral structure of X ⊗2k i 20 References 21

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An elementary method to compute the algebra generated by some given matrices and its dimension

Cited by 11 publications

References 9 publications

On real algebras generated by positive and nonnegative matrices

On real algebras generated by positive and nonnegative matrices

Effective noncommutative Nevanlinna-Pick interpolation in the row ball, and applications

The outer spectral radius and dynamics of completely positive maps

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