Author(s): Alka Marwaha and Rashmi Sehgal Thukral

Abstract: An r-potent matrix ^[9] in M_n (R) is an n×n matrix satisfying E^r=E. An idempotent matrix is an r-potent with r=2. A multiplicative semigroup S in M_n (R) is to be said decomposable (see ^[2]) if there exists a special kind of common invariant subspace called standard invariant subspace for each A∈S. A semi-group S of non-negative r-potent matrices in M_n (R) is known (see ^[5]) to be decomposable if rank (S)>r-1 for all S in S. Our contributions in this paper are as follows: We study the structure of decomposable semi-groups of non-negative r-potent matrices in M_n (R). We reduce these decomposable semi-groups into standard block triangular form wherein the diagonal blocks form constant rank indecomposable semi-groups of non-negative r-potents. Under the special condition of fullness, we obtain a block diagonalization of the decomposable semigroup of non-negative r-potent matrices. Lastly, we shall illustrate the complete structure of the maximal indecomposable semigroup of 2-potents (idempotent) with constant rank one.

Pages: 215-225 | Views: 307 | Downloads: 18

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