In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of co...In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.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
文摘In this paper we first consider the existence and the general form of solution to the following generalized inverse eigenvalue problem(GIEP): given a set of n-dimension complex vectors {x j}m j=1 and a set of complex numbers {λ j}m j=1, find two n×n centrohermitian matrices A,B such that {x j}m j=1 and {λ j}m j=1 are the generalized eigenvectors and generalized eigenvalues of Ax=λBx, respectively. We then discuss the optimal approximation problem for the GIEP. More concretely, given two arbitrary matrices, , ∈C n×n, we find two matrices A and B such that the matrix (A*,B*) is closest to (,) in the Frobenius norm, where the matrix (A*,B*) is the solution to the GIEP. We show that the expression of the solution of the optimal approximation is unique and derive the expression for it.
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.
