This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition f...This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi’s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.展开更多
Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ...Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ajj｜≤rirj｝. The resultre- duces the num berofovals in originalBrauer'stheorem in m any cases. Eigenval- ues(and associated eigenvectors) thatlocate in theboundary ofΩ～ arediscussed.展开更多
In this paper we present some new absolute and relative perturbation bounds for the eigenvalue for arbitrary matrices, which improves some recent results. The eigenvalue inclusion region is also discussed.
文摘This paper obtains a necessary and sufficient condition for an irreducible complex matrix whose comparison matrix is a singular M-matrix to be singular. This is used to establish a necessary and sufficient condition for a boundary point of Brualdi’s inclusion region of the eigenvalues of an irreducible complex matrix to be an eigenvalue.
文摘Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ajj｜≤rirj｝. The resultre- duces the num berofovals in originalBrauer'stheorem in m any cases. Eigenval- ues(and associated eigenvectors) thatlocate in theboundary ofΩ～ arediscussed.
文摘In this paper we present some new absolute and relative perturbation bounds for the eigenvalue for arbitrary matrices, which improves some recent results. The eigenvalue inclusion region is also discussed.
