it is proven that if and only if the weight vector of the neural network is equal to the generalized eigenvector corresponding to the largest generalized eigenvalue of a Hermitian matrix pencil,the proposed algorithm reaches to convergence status.By using the deterministic discretetime(DDT)method,...
The generalized eigenvector problem (GEP) for Hermitian Toeplitz matrices has been studied by transforming it into an eigenvector problem for a centrohermitian matrix. It is found that as long as its real eigenvalues have a geometric multiplicity of 1, the associated eigenvectors of a centrohermiti...
In physics, we sometimes can be obsessed with Hermitian matrices (because of their physical significance in quantum mechanics) which always diagonalize to nice real valued numbers. And now that you mention it, I now recall that even for Hermitian matrices, my office mate was using Lanczos to ...
If a matrix A is Hermitian, or real and symmetric, all its eigenvalues are real. The relation between Hermitian matrices and their eigenvalue is further deepened also for positive-definite, semipositive-definite, and their negative counterparts, as the resulting eigenvalues would be (λ>0), (λ...
where is a diagonal matrix, so it must be true that is also diagonal. In particular, if is a symmetric matrix, then the left and right eigenvectors are simply each other's transpose, and if is a self-adjoint matrix (i.e., it is Hermitian), then the left and right eigenvectors are ...
As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. 对于埃尔米特矩阵,关键在于存在至少一个非零向量。 LASER-wikipedia2 The smallest eigenvalue for every Laplacian matrix is 0, and its corresponding eigenvector is [1, 1, . . . , 1]...
A∗, AT, AH, and A† denote the conjugate, transpose, Hermitian transpose, and pseudo inverse of A, respectively. We will use ⊗ for the Kronecker product, ⊙ for the Khatri-Rao (columnwise Kronecker) product [23], I n for a n×n identify matrix, 0m×n for an m×n zero ...
We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We assume that the ...
The matrix equation AXBH=E with SX=XR or PX=sXQ constraint is considered, where S, R are Hermitian idempotent, P, Q are Hermitian involutory, and s=??1. By the eigenvalue decompositions of S, R, the equation AXBH=E with SX=XR constraint is equivalently transformed to an ...
Mehlig, Eigenvector statistics in non-hermitian random matrix ensembles, Phys. Rev. Lett. 81, 3367 (1998).B. Mehlig and J. T. Chalker, Eigenvector correlations in non-Hermitian random matrix ensembles, Ann. Phys. 7 (1998), no. 5-6, 427-436....