Arrowhead operators on a Hilbert space

Marrero, J. Abderramán; Tomeo, V.

doi:10.7153/oam-10-34

Cited by 1 publication

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“…From here on, we take this convention by assuming c i = a i and denoting A ={ a i , b i } n without loss of generality. Some known results are adapted here to the symmetric case.…”

Section: Preliminariesmentioning

confidence: 99%

“…Thus, the matrix inverse A −1 can be factored trivially. Indeed, the complete proof of this result is shown in , where authors prove that every nonsingular arrowhead matrix A ={ a i , b i , c i } n , with

b_{i} \neq 0, i = 1, 2, \dots, n

has what they call a U D L factorization. Theorem is a particular case of this result when a i = c i , for

i = 1, 2, \dots, n

.…”

Section: Preliminariesmentioning

confidence: 99%

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Associating hub‐directed multigraphs to arrowhead matrices

Marrero

Valdés

Villar

2017

Math Methods in App Sciences

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In this paper, we use the arrowhead matrices as a tool to study graph theory. More precisely, we deal with an interesting class of directed multigraphs, the hub‐directed multigraphs. We associate the arrowhead matrices with the adjacency matrices of a class of directed multigraph, and we obtain new properties of the second objects by using properties of the first ones. The hub‐directed multigraphs with potential use in applications are also defined. As main result, we show that a hub‐directed multigraph G(H) with adjacency matrix H∗ is a dominant hub‐directed multigraph if and only if H∗=CE, where C is the adjacency matrix of another directed multigraph and E is the adjacency matrix of a particular elementary dominant hub‐directed pseudo‐graph. Another decomposition of its Gram (arrowhead) matrix A=()H∗TH∗ is also given. Copyright © 2017 John Wiley & Sons, Ltd.

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“…From here on, we take this convention by assuming c i = a i and denoting A ={ a i , b i } n without loss of generality. Some known results are adapted here to the symmetric case.…”

Section: Preliminariesmentioning

confidence: 99%