International Journal of Statistics and Applied Mathematics
2022, Vol. 7, Issue 5, Part B
A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ(R)
Author(s): Alka Marwaha and Rashmi Sehgal Thukral
Abstract: An ݑ-potent matrix in ݑ0ݑ(R) is an ݑ × ݑ matrix satisfying ݐذݑ = ݐ¸. A multiplicative semigroup S in ݑ0ݑ(R) is said to be decomposable if there exists a special kind of common invariant subspace called standard invariant subspace for each ݐ´ ∈ S. A semi-group S of non-negative ݑ-potent matrices in ݑ0ݑ(R) is known to be decomposable if rank (ݑ) > ݑ − 1 for all ݑ in S. Further, a semigroup S in ݑ0ݑ(R) of nonnegative matrices will be called a full semigroup if S has no common zero row and no common zero column. We have studied the structure of maximal semi-groups of non-negative ݑ-potent matrices in ݑ0ݑ(R) under the special condition of fullness. The objectives of this paper are twofold: (1) To find conditions under which semigroups of nonnegative r-potent matrices can be expressed as a direct sum of maximal rank-one indecomposable semigroups of r-potent matrices; and (2) To obtain a canonical representation of maximal indecomposable rank-one semigroups of r-potent matrices which in the light of the above result gives a complete characterization of such semigroups having constant rank.
Alka Marwaha, Rashmi Sehgal Thukral. A canonical representation of maximal, indecomposable semi groups of constant-rank nonnegative r-potent matrices in ݑ0ݑ(R). Int J Stat Appl Math 2022;7(5):103-107. DOI: 10.22271/maths.2022.v7.i5b.886