%0 Journal Article %1 savchenko03 %A Savchenko, S.V. %D 2003 %J Linear Algebra and its Applications %K eigenvalues matrix nonnegative perron-frobenius principal.submatrix %P 257--277 %R 10.1016/S0024-3795(02)00492-5 %T On the Perron roots of principal submatrices of co-order one of irreducible nonnegative matrices %V 361 %X Let $A$ be an irreducible nonnegative matrix, $w$ be any of its indices, and $A-w$ be the principal submatrix of co-order one obtained from $A$ by deleting the $w$th column and row. Denote by $V_ext(A)$ the set of indices $w$ such that $A − w$ has the biggest Perron root (among all the principal submatrices of co-order one of the original matrix $A$). We prove that exactly one Jordan block corresponds to the Perron root $λ(A − w)$ of $A − w$ for every $w V_ext(A)$. If its size is strictly greater than one for some w ∈ Vext(A), then the original matrix A is permutationally similar to a lower Hessenberg matrix with positive entries on the superdiagonal and in the left lower corner (in other words, the digraph D(A) of A has a Hamiltonian circuit and its diameter is one less than its order). In the opposite case for any w ∈ Vext(A), there is a unique path γ = wi p i=0 going through w in D(A) such that (1) A − wi has the biggest Perron root for i = 0,...,p; (2) A − w0 has a right positive Perron eigenvector; (3) A − wp has a left positive Perron eigenvector; (4) A − wi has neither a left nor a right positive Perron eigenvector for i = 1,...,p − 1. Thus, by the spectral criterion for a nonnegative matrix to be irreducible, the submatrices A − w0,...,A − wp combined inherit the property of irreducibility. We also show that A − w is irreducible for every w ∈ Vext(A) if any of the following holds: (1) A is symmetric; (2) every column (row) of A has at least two positive nondiagonal entries; If A is an irreducible tournament matrix, then either A − w is also irreducible for any w ∈ Vext(A) or there exist exactly two indices w_in and w_out in Vext(A) such that A − win and A − wout are reducible. In the last case any other principal submatrix of co-order one is irreducible. This shows that in the general case, a one-vertex-deleted subdigraph with the biggest Perron root need not have the best connectivity properties among all one-vertex-deleted subdigraphs of a given strongly connected digraph D.

@article{savchenko03, abstract = {Let $A$ be an irreducible nonnegative matrix, $w$ be any of its indices, and $A-w$ be the principal submatrix of co-order one obtained from $A$ by deleting the $w$th column and row. Denote by $V_ext(A)$ the set of indices $w$ such that $A − w$ has the biggest Perron root (among all the principal submatrices of co-order one of the original matrix $A$). We prove that exactly one Jordan block corresponds to the Perron root $λ(A − w)$ of $A − w$ for every $w \in V_ext(A)$. If its size is strictly greater than one for some w ∈ Vext(A), then the original matrix A is permutationally similar to a lower Hessenberg matrix with positive entries on the superdiagonal and in the left lower corner (in other words, the digraph D(A) of A has a Hamiltonian circuit and its diameter is one less than its order). In the opposite case for any w ∈ Vext(A), there is a unique path γ = {wi} p i=0 going through w in D(A) such that (1) A − wi has the biggest Perron root for i = 0,...,p; (2) A − w0 has a right positive Perron eigenvector; (3) A − wp has a left positive Perron eigenvector; (4) A − wi has neither a left nor a right positive Perron eigenvector for i = 1,...,p − 1. Thus, by the spectral criterion for a nonnegative matrix to be irreducible, the submatrices A − w0,...,A − wp combined inherit the property of irreducibility. We also show that A − w is irreducible for every w ∈ Vext(A) if any of the following holds: (1) A is symmetric; (2) every column (row) of A has at least two positive nondiagonal entries; If A is an irreducible tournament matrix, then either A − w is also irreducible for any w ∈ Vext(A) or there exist exactly two indices w_in and w_out in Vext(A) such that A − win and A − wout are reducible. In the last case any other principal submatrix of co-order one is irreducible. This shows that in the general case, a one-vertex-deleted subdigraph with the biggest Perron root need not have the best connectivity properties among all one-vertex-deleted subdigraphs of a given strongly connected digraph D.}, added-at = {2017-01-31T11:39:43.000+0100}, author = {Savchenko, S.V.}, biburl = {https://www.bibsonomy.org/bibtex/2dad16b39ae8cda097d54446a95cbe47d/ytyoun}, doi = {10.1016/S0024-3795(02)00492-5}, interhash = {2838af4af27b7ae6e8bba4c6cfbf4077}, intrahash = {dad16b39ae8cda097d54446a95cbe47d}, issn = {0024-3795}, journal = {Linear Algebra and its Applications}, keywords = {eigenvalues matrix nonnegative perron-frobenius principal.submatrix}, pages = {257--277}, timestamp = {2017-11-20T02:14:49.000+0100}, title = {On the {Perron} roots of principal submatrices of co-order one of irreducible nonnegative matrices}, volume = 361, year = 2003 }