Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Cross-Dimensional Linear Systems   Semi-tensor product(STP) or matrix (M-) product of matrices turns the set of
matrices with arbitrary dimensions into a monoid $({\cal M},\ltimes)$. A matrix
(M-) addition is defined over subsets of a partition of ${\cal M}$, and a
matrix (M-) equivalence is proposed. Eventually, some quotient spaces are
obtained as vector spaces of matrices. Furthermore, a set of formal polynomials
is constructed, which makes the quotient space of $({\cal M},\ltimes)$, denoted
by $(\Sigma, \ltimes)$, a vector space and a monoid.
  Similarly, a vector addition (V-addition) and a vector equivalence
(V-equivalence) are defined on ${\cal V}$, the set of vectors of arbitrary
dimensions. Then the quotient space of vectors, $\Omega$, is also obtained as a
vector space.
  The action of monoid $({\cal M},\ltimes)$ on ${\cal V}$ (or $(\Sigma,
\ltimes)$ on $\Omega$) is defined as a vector (V-) product, which becomes a
pseudo-dynamic system, called the cross-dimensional linear system (CDLS). Both
the discrete time and the continuous time CDLSs have been investigated. For
certain time-invariant case, the solutions (trajectories) are presented.
Furthermore, the corresponding cross-dimensional linear control systems are
also proposed and the controllability and observability are discussed.
  Both M-product and V-product are generalizations of the conventional matrix
product, that is, when the dimension matching condition required by the
conventional matrix product is satisfied they coincide with the conventional
matrix product. Both M-addition and V-addition are generalizations of
conventional matrix addition. Hence, the dynamics discussed in this paper is a
generalization of conventional linear system theory.

22
The categories of this paper are: chao-dyn
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
36
The categories of this paper are: cmp-lg
24
The categories of this paper are: dg-ga