The Wielandt-Hoffman theorem of the real symmetric matrix is extended into a plural matrix. On the basis of it, a similar theory about the trace of a matrix for the arithmetic mean, geometric mean inequality, Holder i...The Wielandt-Hoffman theorem of the real symmetric matrix is extended into a plural matrix. On the basis of it, a similar theory about the trace of a matrix for the arithmetic mean, geometric mean inequality, Holder inequality and Minkowski inequality is proved.展开更多
This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformat...This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.展开更多
The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidski...The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidskiǐ inequalities.An elementary proof of the latter for hyperbolic polynomials is given.This proof follows an idea from H.Weinberger and is free from representation theory and Schubert calculus arguments,as well as from hyperbolic partial differential equations theory.展开更多
By virtue of the two-layer picture of Pfatfian pair Hall state, a qubit representation of topological degeneracy for quasiholes excitation is displayed. The non-Abelian feature of states can be manifested readily by t...By virtue of the two-layer picture of Pfatfian pair Hall state, a qubit representation of topological degeneracy for quasiholes excitation is displayed. The non-Abelian feature of states can be manifested readily by the new wave functions. The virtue of this approach is that one does not need to find the equalities of Pfalfians, which is a so tedious task as exemplified for the case of six quasiholes. Then the braiding matrices are also constructed readily by just permutating single-qubit states, which are unitary and hermite.展开更多
文摘The Wielandt-Hoffman theorem of the real symmetric matrix is extended into a plural matrix. On the basis of it, a similar theory about the trace of a matrix for the arithmetic mean, geometric mean inequality, Holder inequality and Minkowski inequality is proved.
基金This work is supported by the National Natural Science Foundation of China under Grant Nos. 10771022 and 10571012, Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China under Grant No. 890 [2008], and Major Foundation of Educational Committee of Hunan Province under Grant No. 09A002 [2009] Portuguese Foundation for Science and Technology （FCT） through the Research Programme POCTI, respectively.
文摘This paper concerns the reconstruction of an hermitian Toeplitz matrix with prescribed eigenpairs. Based on the fact that every centrohermitian matrix can be reduced to a real matrix by a simple similarity transformation, the authors first consider the eigenstructure of hermitian Toeplitz matrices and then discuss a related reconstruction problem. The authors show that the dimension of the subspace of hermitian Toeplitz matrices with two given eigenvectors is at least two and independent of the size of the matrix, and the solution of the reconstruction problem of an hermitian Toeplitz matrix with two given eigenpairs is unique.
文摘The roots of hyperbolic polynomials satisfy the linear inequalities that were previously established for the eigenvalues of Hermitian matrices,after a conjecture by A.Horn.Among them are the so-called Weyl and Lidskiǐ inequalities.An elementary proof of the latter for hyperbolic polynomials is given.This proof follows an idea from H.Weinberger and is free from representation theory and Schubert calculus arguments,as well as from hyperbolic partial differential equations theory.
基金Supported by National Natural Science Foundation of China under Grant Nos.11005002 and 11005003New Century Excellent Talent of M.O.E(NCET-11-0937)Sponsoring Program of Excellent Younger Teachers in universities in Henan Province of China under Grant No.2010GGJS-181
文摘By virtue of the two-layer picture of Pfatfian pair Hall state, a qubit representation of topological degeneracy for quasiholes excitation is displayed. The non-Abelian feature of states can be manifested readily by the new wave functions. The virtue of this approach is that one does not need to find the equalities of Pfalfians, which is a so tedious task as exemplified for the case of six quasiholes. Then the braiding matrices are also constructed readily by just permutating single-qubit states, which are unitary and hermite.
