Inertia and biclique decompositions of joins of graphs

“…The inertia of rank 1 Hermitian perturbation of Hermitian matrix was considered in [17]. Now setting A = αI m in Theorem 2.3, we obtain the following result on the rank and inertia of a Hermitian matrix with a Hermitian perturbation of arbitrary rank.…”

Section: Substituting Them Into (22)-(25) Leads To (214)-(217)mentioning

confidence: 93%

“…Equation (2.1) is a standard form of block Hermitian matrix in the investigations of Hermitian matrices and their applications. Equalities and inequalities for the rank and inertia of the block Hermitian matrix in (2.1), as well as various completion problems associated with its rank and inertia were widely studied in the literature; see, e.g., [2]- [5], [8], [11]- [14], [17], [18] and [26]. Also note that the M (X) in (2.1) can be rewritten as…”

mentioning

confidence: 99%

“…Some previous work on the inertias of the block matrices in (2.1) and (3.1) with restrictions to the specified or unspecified entries can be found, e.g., in [8]- [10] and [17]. As a continuation of the work in the previous sections, it is of interest to derive maximal and minimal ranks and inertias r( B ± X * AX ) = r(B) ± r(X * AX), i ± ( B ± X * AX ) = i ± (B) ± i ± (X * AX).…”

mentioning

confidence: 99%

See 1 more Smart Citation

Completing block Hermitian matrices with maximal and minimal ranks and inertias

Tian¹

2010

ELA

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“…The proof can then be finished by comparing (8) and (9), and by using the fact that ∼ is a rank-preserving relation.…”

Section: Displacement Structuresmentioning

confidence: 99%

Structures preserved by matrix inversion

Delvaux¹,

Barel²

2006

SIAM J. Matrix Anal. & Appl.

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In this paper we investigate some matrix structures on C n×n that have a good behaviour under matrix inversion. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, also the inverse matrix must have a low rank submatrix, which we can explicitly determine. This allows us to generalize a theorem due to Fiedler. The generalization consists in the fact that our rank structures may have their own shift matrix Λ k ∈ C m×m , for suitable m, with Fiedler's theorem corresponding to the limiting cases Λ k → 0 and Λ k → ∞I.Keywords : displacement structures, Hermitian plus low rank, rank structures, lower semiseparable (plus diagonal) matrices, matrix inversion. AMS(MOS) Classification : Primary : 15A09, Secondary : 15A03,65F05. Structures preserved by matrix inversionSteven Delvaux * , Marc Van Barel * 24th December 2004Abstract In this paper we investigate some matrix structures on C n×n that have a good behaviour under matrix inversion. The first type of structure is closely related to low displacement rank matrices. Next, we show that for a matrix having a low rank submatrix, also the inverse matrix must have a low rank submatrix, which we can explicitly determine. This allows us to generalize a theorem due to Fiedler. The generalization consists in the fact that our rank structures may have their own shift matrix Λ k ∈ C m×m , for suitable m, with Fiedler's theorem corresponding to the limiting cases Λ k → 0 and Λ k → ∞I.

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Inertia and biclique decompositions of joins of graphs

Cited by 27 publications

References 10 publications

On decompositions of complete hypergraphs

On decompositions of complete hypergraphs

Completing block Hermitian matrices with maximal and minimal ranks and inertias

Structures preserved by matrix inversion

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