A remark on eigen values of signed graph

Prashanth, B.; Naik, Kshipra; Rajanna, K. R.

doi:10.2478/jamsi-2019-0003

Cited by 3 publications

(1 citation statement)

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“…al. [25] introduced some results using eigenvalues of signed graph using the number of vertices. In addition to these applications, stochastic modeling is an interesting and challenging domain of statistics and data science, for more details see Bhuiyan [3].…”

Section: Introductionmentioning

confidence: 99%

Doubly stochastic matrices and the quantum channels

Das

Ahmed

2021

Journal of Applied Mathematics, Statistics and Informatics

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The main object of this paper is to study doubly stochastic matrices with majorization and the Birkhoff theorem. The Perron-Frobenius theorem on eigenvalues is generalized for doubly stochastic matrices. The region of all possible eigenvalues of n-by-n doubly stochastic matrix is the union of regular (n – 1) polygons into the complex plane. This statement is ensured by a famous conjecture known as the Perfect-Mirsky conjecture which is true for n = 1, 2, 3, 4 and untrue for n = 5. We show the extremal eigenvalues of the Perfect-Mirsky regions graphically for n = 1, 2, 3, 4 and identify corresponding doubly stochastic matrices. Bearing in mind the counterexample of Rivard-Mashreghi given in 2007, we introduce a more general counterexample to the conjecture for n = 5. Later, we discuss different types of positive maps relevant to Quantum Channels (QCs) and finally introduce a theorem to determine whether a QCs gives rise to a doubly stochastic matrix or not. This evidence is straightforward and uses the basic tools of matrix theory and functional analysis.

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Section: Introductionmentioning

confidence: 99%