Some properties of matrix partial orderings

Baksalary, Jerzy K.; Pukelsheim, Friedrich; Styan, George P. H.

doi:10.1016/0024-3795(89)90069-4

Cited by 58 publications

(27 citation statements)

References 24 publications

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“…Let a, b ∈ R. Then a is below b under the minus partial order, denoted by a < − b, if a ∈ R (1) and ax = bx, xa = xb for some x ∈ a{1}.…”

Section: Ds Rakićmentioning

confidence: 99%

“…However, we prove it here for completeness. As a < − b, there exists a (1) ∈ a{1} such that aa (1) = ba (1) and a…”

Section: Elamentioning

confidence: 99%

“…Reflexivity and transitivity follows from Theorem 3.3. If a < − b and b < − a, then a{1} ⊆ b{1} and there exists a (1) ∈ a{1} such that aa (1) = ba (1) and a…”

Section: Elamentioning

confidence: 99%

“…For example, when a < − b, we can easily characterize all elements x appearing in (1.1) and the class of all idempotents p and q such that a = pb = bq. The same notion of decomposition enable us to find equivalent conditions for the invariance of ab (1) a under the choices of b (1) ∈ b{1}. Also, it is proven that the conditions a < − b, b{1} ⊆ a{1} and b{1, 2} ⊆ a{1} are equivalent, which is known for matrices, (see [12,17]).…”

mentioning

confidence: 92%

“…We denote by a{1} the set of all generalized inverses of a and by a{1, 2} the set of all reflexive generalized inverses of a. The set of all regular elements of R is denoted by R (1) . Some properties of generalized inverses in a ring was studied in [2,4,15].…”

mentioning

confidence: 99%

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Decomposition of a ring induced by minus partial order

Rakić¹

2012

ELA

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Abstract. The minus partial order linear algebraic methods have proven to be useful in the study of complex matrices. This paper extends study of minus partial orders to general rings. It is shown that the condition a < − b, where < − is the minus partial order, defines two triples of orthogonal idempotents, and thus, a decomposition of a ring into a direct sum of abelian groups. Hence, several well-known results concerning minus partial order on real and complex matrices are generalized. Analogous decompositions for rectangular matrices over a ring and for Banach space operators are obtained. The equivalent conditions for the invariance of ab (1) a under the choices of b (1) are also obtained. An original inspiration for this work came from the study of minus partial order on complex matrices and from linear algebra methods.

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“…Let a, b ∈ R. Then a is below b under the minus partial order, denoted by a < − b, if a ∈ R (1) and ax = bx, xa = xb for some x ∈ a{1}.…”

Section: Ds Rakićmentioning

confidence: 99%

“…However, we prove it here for completeness. As a < − b, there exists a (1) ∈ a{1} such that aa (1) = ba (1) and a…”

Section: Elamentioning

confidence: 99%

“…Reflexivity and transitivity follows from Theorem 3.3. If a < − b and b < − a, then a{1} ⊆ b{1} and there exists a (1) ∈ a{1} such that aa (1) = ba (1) and a…”

Section: Elamentioning

confidence: 99%

mentioning

confidence: 92%

mentioning

confidence: 99%

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