3 research outputs found
On Potential Equations of Finite Games
In this paper, some new criteria for detecting whether a finite game is
potential are proposed by solving potential equations. The verification
equations with the minimal number for checking a potential game are obtained
for the first time. Some connections between the potential equations and the
existing characterizations of potential games are established. It is revealed
that a finite game is potential if and only if its every bi-matrix sub-game is
potential
Topologies on Quotient Space of Matrices via Semi-tensor Product
An equivalence of matrices via semi-tensor product (STP) is proposed. Using
this equivalence, the quotient space is obtained. Parallel and sequential
arrangements of the natural projection on different shapes of matrices leads to
the product topology and quotient topology respectively. Then the Frobenious
inner product of matrices is extended to equivalence classes, which produces a
metric on the quotient space. This metric leads to a metric topology. A
comparison for these three topologies is presented. Some topological properties
are revealed.Comment: 8 pages, 2 figure
On Equivalence of Matrices
A new matrix product, called the semi-tensor product (STP), is briefly
reviewed. The STP extends the classical matrix product to two arbitrary
matrices. Under STP the set of matrices becomes a monoid (semi-group with
identity). Some related structures and properties are investigated. Then the
generalized matrix addition is also introduced, which extends the classical
matrix addition to a class of two matrices with different dimensions.
Motivated by STP of matrices, two kinds of equivalences of matrices
(including vectors) are introduced, which are called matrix equivalence
(M-equivalence) and vector equivalence (V-equivalence) respectively. The
lattice structure has been established for each equivalence. Under each
equivalence, the corresponding quotient space becomes a vector space. Under
M-equivalence, many algebraic, geometric, and analytic structures have been
posed to the quotient space, which include (i) lattice structure; (ii) inner
product and norm (distance); (iii) topology; (iv) a fiber bundle structure,
called the discrete bundle; (v) bundled differential manifold; (vi) bundled Lie
group and Lie algebra. Under V-equivalence, vectors of different dimensions
form a vector space , and a matrix of arbitrary dimension is
considered as an operator (linear mapping) on . When is a bounded
operator (not necessarily square but includes square matrices as a special
case), the generalized characteristic function, eigenvalue and eigenvector etc.
are defined.
In one word, this new matrix theory overcomes the dimensional barrier in
certain sense. It provides much more freedom for using matrix approach to
practical problems