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Certain additive decompositions in a noncommutative ring

Huanyin Chen, Marjan Sheibani, Rahman Bahmani (2022)

Czechoslovak Mathematical Journal

We determine when an element in a noncommutative ring is the sum of an idempotent and a radical element that commute. We prove that a 2 × 2 matrix A over a projective-free ring R is strongly J -clean if and only if A J ( M 2 ( R ) ) , or I 2 - A J ( M 2 ( R ) ) , or A is similar to 0 λ 1 μ , where λ J ( R ) , μ 1 + J ( R ) , and the equation x 2 - x μ - λ = 0 has a root in J ( R ) and a root in 1 + J ( R ) . We further prove that f ( x ) R [ [ x ] ] is strongly J -clean if f ( 0 ) R be optimally J -clean.

Characterization and properties of (Pσ, Q) symmetric and co-symmetric matrices

William F. Trench (2014)

Special Matrices

Let P ∈ ℂmxm and Q ∈ ℂn×n be invertible matrices partitioned as P = [P0 P1 · · · Pk−1] and Q = [Q0 Q1 · · · Qk−1], with P ℓ ∈ ℂm×mℓ and Qℓ ∈ ℂn×nℓ , 0 ≤ ℓ ≤ k − 1. Partition P−1 and Q−1 as [...] where P̂ℓ ∈ ℂmℓ ×m, Q̂ℓ ∈ ℂnℓ×n , P̂ℓPm = δℓmImℓ , and Q̂ℓQm = δℓmInℓ , 0 ≤ ℓ, m ≤ k − 1. Let Zk = {0, 1, . . . , k − 1}. We study matrices A = [...] Pσ(ℓ)FℓQℓ and B = [...] QℓGℓPσ(ℓ), where σ : Zk → Zk. Special cases: A = [...] and B = [...] , where Aℓ ∈ ℂd1×d2 and Bℓ ∈ ℂd2×d1, 0 ≤ ℓ ≤ k − 1.

Computing generalized inverse systems using matrix pencil methods

Andras Varga (2001)

International Journal of Applied Mathematics and Computer Science

We address the numerically reliable computation of generalized inverses of rational matrices in descriptor state-space representation. We put particular emphasis on two classes of inverses: the weak generalized inverse and the Moore-Penrose pseudoinverse. By combining the underlying computational techniques, other types of inverses of rational matrices can be computed as well. The main computational ingredient to determine generalized inverses is the orthogonal reduction of the system matrix pencil...

Constructions for type I trees with nonisomorphic Perron branches

Stephen J. Kirkland (1999)

Czechoslovak Mathematical Journal

A tree is classified as being type I provided that there are two or more Perron branches at its characteristic vertex. The question arises as to how one might construct such a tree in which the Perron branches at the characteristic vertex are not isomorphic. Motivated by an example of Grone and Merris, we produce a large class of such trees, and show how to construct others from them. We also investigate some of the properties of a subclass of these trees. Throughout, we exploit connections between...

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