Uniqueness of matrix square roots and an application

Johnson, Charles R.; Okubo, Kimihiro; Reams, Robert

doi:10.1016/s0024-3795(00)00243-3

Cited by 45 publications

(26 citation statements)

References 6 publications

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“…In [3], [4] and [5], by using techniques of linear algebra, the same was proved for p = 2 and H = M n (C). In [8] the results extended to the case of C * -algebras and each real number ω ∈ [1, ∞).…”

Section: Theorem 1 If T Is a Bounded Linear Operator On A Complex Hilmentioning

confidence: 64%

Roots of C*-algebra elements under numerical range condition

Abdollahi

Heydari

2009

Rend. Circ. Mat. Palermo

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Let A be a C*-algebra with unit 1. For each a ∈ A, the C*-algebra numerical range is defined by V (a) = {ϕ(a) : ϕ ∈ A * ,ϕ ≥ 0,ϕ(1) = 1}. In a 2003 paper Li, Rodman and Spitkovsky have found the ω-th roots of elements in C*-algebra under a numerical range condition, when ω ∈ [1, ∞). In this paper, we will give a short proof of the above result in the case of ω is a positive integer number. We also give a simple proof for ω-th root of an element a ∈ A, when ω ∈ [1, ∞) and V (a) ∩ {z ∈ C : z ≤ 0} = ∅.Let A be a C*-algebra with unit 1 and let S be the state space of A, i.e. S = {ϕ ∈ A * : ϕ ≥ 0,ϕ(1) = 1}. For each a ∈ A, the C*-algebra numerical range is defined byIt is well known that V (a) is non empty, compact and convex subset of the complex plane, V (α1 + βa) = α + βV (a) for a ∈ A and α, β ∈ C, and if z ∈ V (a), |z| ≤ a (For further details see [2]).

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Section: Theorem 1 If T Is a Bounded Linear Operator On A Complex Hilmentioning

confidence: 64%

Roots of C*-algebra elements under numerical range condition

Abdollahi

Heydari

2009

Rend. Circ. Mat. Palermo

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“…Over the field of complex numbers or the reals, matrix roots are a well-studied and still up-to-date topic of linear algebra [11,7,17]. But results from that field of research do generally not apply to Boolean matrices.…”

Section: Related Work-related Questionsmentioning

confidence: 99%

The Complexity of Boolean Matrix Root Computation

Kutz¹

2003

Lecture Notes in Computer Science

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Abstract. We show that finding roots of Boolean matrices is an NPhard problem. This answers a twenty year old question from semigroup theory. Interpreting Boolean matrices as directed graphs, we further reveal a connection between Boolean matrix roots and graph isomorphism, which leads to a proof that for a certain subclass of Boolean matrices related to subdivision digraphs, root finding is of the same complexity as the graph-isomorphism problem.

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“…The problem of matrix square root is widely encountered in many scientific areas [1][2][3][4][5][6]. Due to its fundamental roles, much effort has been directed toward the solving algorithms of matrix square root [1][2][3][4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…Due to its fundamental roles, much effort has been directed toward the solving algorithms of matrix square root [1][2][3][4][5][6]. Being one of the most useful methods, Newton iteration [1,3] has been investigated for matrix square root finding, owing to its good properties of convergence and stability.…”

Section: Introductionmentioning

confidence: 99%

“…Generally speaking, the methods reported in [1][2][3][4][5][6][7][8] (including the ones mentioned earlier) are related to the numerical algorithms (which may be less efficient enough for large-scale and/or online problem solving, due to their serial-processing nature). Being another important type of solution, many parallel-processing computational schemes, including various dynamic recurrent neural networks (RNN) solvers, have been developed, analyzed, and implemented on specific architectures [9][10][11][12][13].…”

Section: Introductionmentioning

confidence: 99%

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Zhang neural network and its application to Newton iteration for matrix square root estimation

Zhang

Yang

Cai

et al. 2010

Neural Comput & Applic

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A special class of recurrent neural networks (RNN) has recently been proposed by Zhang et al. for solving online time-varying matrix problems. Being different from conventional gradient-based neural networks (GNN), such RNN (termed specifically as Zhang neural networks, ZNN) are designed based on matrix-valued error functions, instead of scalar-valued norm-based energy functions. In this paper, we generalize and further investigate the ZNN model for time-varying matrix square root finding. For the purpose of possible hardware (e.g., digital circuit) realization, a discrete-time ZNN model is constructed and developed, which incorporates Newton iteration as a special case. Besides, to obtain an appropriate step-size value (in each iteration), a line-search algorithm is employed for the proposed discrete-time ZNN model. Computer-simulation results substantiate the effectiveness of the proposed ZNN model aided with a line-search algorithm, in addition to the connection and explanation to Newton iteration for matrix square root finding.

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Uniqueness of matrix square roots and an application

Cited by 45 publications

References 6 publications

Roots of C*-algebra elements under numerical range condition

Roots of C*-algebra elements under numerical range condition

The Complexity of Boolean Matrix Root Computation

Zhang neural network and its application to Newton iteration for matrix square root estimation

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