Linear maps preserving the higher numerical ranges of tensor product of matrices

We study linear maps preserving the higher numerical ranges of tensor product of matrices

Linear maps preserving numerical radius of tensor product of matrices

We determine the structure of linear maps on the tensor product of matrices which preserve the numerical range or numerical radius.Comment: 10 page

Interplay Between Game Theory and Control Theory

An interrelationship between Game Theory and Control Theory is seeked. In this respect two aspects of this relationship are brought up. To establish the direct relationship Control Based Games and to establish the inverse relationship Game Based Control are discussed. In the attempt to establish the direct relationship Control Based Boolean Networks are discussed with the novel technique of application of Semi Tensor Product of matrices in Differential Calculus. For the inverse relationship H Infinity robust optimal control has been discussed with the help of Dynamic Programming and Pontryagin Minimization Principle.Comment: 29 pages no figure

Observability of Boolean Networks via Set Controllability Approach

The controllability and observability of Boolean control network(BCN) are two fundamental properties. But the verification of latter is much harder than the former. This paper considers the observability of BCN via controllability. First, the set controllability is proposed, and the necessary and sufficient condition is obtained. Then a technique is developed to convert the observability into an equivalent set controllability problem. Using the result for set controllability, the necessary and sufficient condition is also obtained for the observability of BCN.Comment: 6 page

On Skew-Symmetric Games

By resorting to the vector space structure of finite games, skew-symmetric games (SSGs) are proposed and investigated as a natural subspace of finite games. First of all, for two player games, it is shown that the skew-symmetric games form an orthogonal complement of the symmetric games. Then for a general SSG its linear representation is given, which can be used to verify whether a finite game is skew-symmetric. Furthermore, some properties of SSGs are also obtained in the light of its vector subspace structure. Finally, a symmetry-based decomposition of finite games is proposed, which consists of three mutually orthogonal subspaces: symmetric subspace, skew-symmetric subspace and asymmetric subspace. An illustrative example is presented to demonstrate this decomposition.Comment: 31 pages,9 table

On Coset Weighted Potential Game

In this paper we first define a new kind of potential games, called coset weighted potential game, which is a generalized form of weighted potential game. Using semi-tensor product of matrices, an algebraic method is provided to verify whether a finite game is a coset weighted potential game, and a simple formula is obtained to calculate the corresponding potential function. Then some properties of coset weighted potential games are revealed. Finally, by resorting to the vector space structure of finite games, a new orthogonal decomposition based on coset weights is proposed, the corresponding geometric and algebraic expressions of all the subspaces are given by providing their bases.Comment: 10 pages,1 figur

Expressing a Tensor Permutation Matrix p⊗np^{\otimes n} in Terms of the Generalized Gell-Mann Matrices

We have shown how to express a tensor permutation matrix $p^{\otimes n}$ as a linear combination of the tensor products of the $p\times p$-Gell-Mann matrices. We have given the expression of a tensor permutation matrix $2\otimes 2 \otimes 2$ as a linear combination of the tensor products of the Pauli matrices.Comment: 15 pages, v2 minor grammatical changes and acknoledgemen

Approximation of The Constrained Joint Spectral Radius via Algebraic Lifting

This paper studies the constrained switching (linear) system which is a discrete-time switched linear system whose switching sequences are constrained by a deterministic finite automaton. The stability of a constrained switching system is characterized by its constrained joint spectral radius that is known to be difficult to compute or approximate. Using the semi-tensor product of matrices, the matrix-form expression of a constrained switching system is shown to be equivalent to that of a lifted arbitrary switching system. Then the constrained joint/generalized spectral radius of a constrained switching system is proved to be equal to the joint/generalized spectral radius of its lifted arbitrary switching system which can be approximated by off-the-shelf algorithms

A Note On Orthogonal Decomposition of Finite Games

Various decomposition of finite games have been proposed. The inner product of vectors plays a key role in the decomposition of finite games. This paper considers the effect of different inner products on the orthogonal decomposition of finite games. We find that only when the compatible condition is satisfied, a common decomposition can be induced by the standard inner product and the weighted inner product. To explain the result, we studied the existing decompositions, including potential based decomposition, zero-sum based decomposition, and symmetry based decomposition.Comment: 4 pages, 2019 IEEE CDC conference paper (submitted

Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

We have expressed the tensor commutation matrix n\otimes n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3\otimes 2 and 2\otimes 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.Comment: 14 pages, Submitted v2:no changes in the body of the paper, just minor grammatical changes in the abstract in replace-for