We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenva...We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .展开更多
The duality of left and right eigenvectors underpins the comprehensive understanding of many physical phenomena.In Hermitian systems,left and right eigenvectors are simply Hermitian-conjugate pairs.In contrast,non-Her...The duality of left and right eigenvectors underpins the comprehensive understanding of many physical phenomena.In Hermitian systems,left and right eigenvectors are simply Hermitian-conjugate pairs.In contrast,non-Hermitian eigenstates have left and right eigenvectors that are distinct from each other.However,despite the tremendous interest in non-Hermitian physics in recent years,the roles of non-Hermitian left eigenvectors(LEVs)are still inadequately explored.Their physical consequences and observable effects remain elusive,so much so that LEVs seem largely like objects of primarily mathematical purpose.In this study,we present a method based on the non-Hermitian Green’s function for directly retrieving both LEVs and right eigenvectors(REVs)from experimentally measured steady-state responses.We validate the effectiveness of this approach in two separate acoustic experiments:one characterizes the non-Hermitian Berry phase,and the other measures extended topological modes.Our results not only unambiguously demonstrate observable effects related to non-Hermitian LEVs but also highlight the under-appreciated role of LEVs in non-Hermitian phenomena.展开更多
文摘We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .
基金supported by the National Key R&D Program(No.2022YFA1404400)the Hong Kong Research Grants Council(Nos.RFS2223-2S01 and 12301822)the Hong Kong Baptist University(Nos.RC-RSRG/23-24/SCI/01 and RC-SFCRG/23-24/R2/SCI/12).
文摘The duality of left and right eigenvectors underpins the comprehensive understanding of many physical phenomena.In Hermitian systems,left and right eigenvectors are simply Hermitian-conjugate pairs.In contrast,non-Hermitian eigenstates have left and right eigenvectors that are distinct from each other.However,despite the tremendous interest in non-Hermitian physics in recent years,the roles of non-Hermitian left eigenvectors(LEVs)are still inadequately explored.Their physical consequences and observable effects remain elusive,so much so that LEVs seem largely like objects of primarily mathematical purpose.In this study,we present a method based on the non-Hermitian Green’s function for directly retrieving both LEVs and right eigenvectors(REVs)from experimentally measured steady-state responses.We validate the effectiveness of this approach in two separate acoustic experiments:one characterizes the non-Hermitian Berry phase,and the other measures extended topological modes.Our results not only unambiguously demonstrate observable effects related to non-Hermitian LEVs but also highlight the under-appreciated role of LEVs in non-Hermitian phenomena.
