In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran's theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157-169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyze successors, predecessors and maximal elements under the diamond order.

5For a {k}-involutory matrix R ∈ C n×n (that is, R k = I n ) and s ∈ {0, 1, 2, 3, . . . },this paper, a matrix group corresponding to a fixed {R, s + 1, k}-potent matrix is 8 explicitly constructed and properties of this group are derived and investigated. This 9 constructed group is then reconciled with the classical matrix group G A that is 10 associated with a generalized group invertible matrix A.

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The inverse eigenvalue problem and the associated optimal approximation problem for Hermitian reflexive matrices with respect to a normal {k +1}-potent matrix are considered. First, we study the existence of the solutions of the associated inverse eigenvalue problem and present an explicit form for them. Then, when such a solution exists, an expression for the solution to the corresponding optimal approximation problem is obtained.

Centro-invertible matrices were introduced by R.S. Wikramaratna in 2008. From an involutory matrix, we introduce generalized centro-invertible matrices and apply them to the modular arithmetic case. Specifically, algorithms for image blurring/deblurring are designed by means of generalized centro-invertible matrices. In addition, we establish that every pair of sets of generalized centro-invertible matrices corresponding to two fixed involutory matrices have the same number of elements.