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Showing papers on "Spectrum of a matrix published in 1978"


Journal ArticleDOI
TL;DR: In this article, it was shown that if the semiclassical estimate is a bound for some moment of the negative eigenvalues (as is known in some cases in one-dimensional spaces), then the same bounds are also bounds for all higher moments.

140 citations


Journal ArticleDOI
TL;DR: In this paper, a simple re-derivation of the Kota and Potbhare result is presented, which corrects the error in the work of Kota-Potbhare.
Abstract: A recently published Letter by Kota and Potbhare (1977) obtains the averaged spectrum of a large symmetric random matrix each element of which has a finite mean: their results disagree with two recent calculations which predict that under certain circumstances a single isolated eigenvalue splits off from the continuous semicircular distribution of eigenvalues associated with the random part of the matrix. This letter offers a simple re-derivation of this result and corrects the error in the work of Kota and Potbhare.

33 citations


Journal ArticleDOI
Jane Cullum1
TL;DR: A heuristic argument and supporting numerical results are given to demonstrate that a block Lanczos procedure can be used to compute simultaneously a few of the algebraically largest and smallest eigenvalues and a corresponding eigenspace of a large, sparse, symmetric matrixA.
Abstract: A heuristic argument and supporting numerical results are given to demonstrate that a block Lanczos procedure can be used to compute simultaneously a few of the algebraically largest and smallest eigenvalues and a corresponding eigenspace of a large, sparse, symmetric matrixA. This block procedure can be used, for example, to compute appropriate parameters for iterative schemes used in solving the equationAx=b. Moreover, if there exists an efficient method for repeatedly solving the equation (A−σI)X=B, this procedure can be used to determine the interior eigenvalues (and corresponding eigenvectors) ofA closest to σ.

31 citations


Journal ArticleDOI
TL;DR: In this article, the behavior of the determinant of the scattering matrix as a function on the spectrum of the unperturbed operator was studied for a large class of scattering systems, where the variation of this determinant was related to the number of eigenvalues due to perturbation.

27 citations


Journal ArticleDOI
TL;DR: In this paper, the vector initial value problem with stable eigenvalues and zero eigen values having simple elementary divisors is considered and the unique limiting solution is determined when the reduced problem is solvable and a uniform asymptotic expansion for the solution on finite t intervals.
Abstract: Consider the vector initial value problem with for a singular matrix F(t) of constant rank with stable eigenvalues and zero eigenvalues having simple elementary divisors. This paper shows how to determine the unique limiting solution when the reduced problem is solvable and how to obtain a uniform asymptotic expansion for the solution on finite t intervals.

20 citations


01 Nov 1978
TL;DR: SRRIT is a FORTRAN program to calculate an approximate orthonormal basis for a dominant invariant subspace of a real matrix A such that the equation AQ = QT is satisfied up to a tolerence specified by the user.
Abstract: : SRRIT is a FORTRAN program to calculate an approximate orthonormal basis for a dominant invariant subspace of a real matrix A. Specifically, given an integer m, SRRIT attempts to compute a matrix Q with m orthonormal columns and real quasi-triangular matrix T of order m such that the equation AQ = QT is satisfied up to a tolerence specified by the user. The eigenvalues of T are approximations to the m largest eigenvalues of A, and the columns of Q span the invariant subspace corresponding to those eigenvalues. SRRIT references A only through a user provided subroutine to form the product AQ; hence it is suitable for large sparse problems. (Author)

19 citations


Journal ArticleDOI
TL;DR: Various methods of approximating the eigenvalues and invariant subspaces of nonself-adjoint differential and integral operators are unified in a general theory.

13 citations


Journal ArticleDOI
TL;DR: An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given in this paper, which leads to estimates of the differences in eigenvalues and the number of points of degeneracy.
Abstract: An elementary and self-contained account of analytic Jordan decomposition of matrix-valued analytic functions is given. An integral representation for their eigenvalues is obtained. This leads to estimates of the differences in eigenvalues and the number of points of degeneracy.

10 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the W. K. B. eigenvalues and eigenfunctions satisfy the Hellmann-Feynman theorem and the virial theorem.
Abstract: It is shown that the W. K. B. eigenvalues and eigenfunctions satisfy the Hellmann-Feynman theorem and the virial theorem. As a consequence the scaling behaviour of the exact eigenvalues with respect to the para­meters in the Hamiltonian is retained in the W. K. B. approximation for the polynomial potentials. Hypervirial relations of certain lower orders are also satisfied by the W. K. B. approximants.

8 citations


Journal ArticleDOI
TL;DR: In this article, the Moore-Penrose inverse of such a "retrocirculant" was determined and the nonzero eigenvalues of the inverse were the reciprocals of the non zero eigen values of the retrocirculant.

7 citations


Journal ArticleDOI
TL;DR: In this article, the basis of a many-phonon system was constructed in the nuclear collective model using the projection-operator method and the proof of the completeness of this basis was given and the matrices of the generators of the group SO(5) are obtained in this basis.
Abstract: The basis of a many-phonon system is constructed in the nuclear collective model using the projection-operator method. The proof of the completeness of this basis is given and the matrices of the generators of the group SO(5) are obtained in this basis. The matrix of the additional integral of motion Omega is calculated. This matrix has a non-degenerate spectrum of eigenvalues omega i which can be used as the missing quantum number.

Journal ArticleDOI
TL;DR: In this article, conditions called condition numbers are computed to measure the sensitivity of the eigenvalues of a matrix and numerical results are given for a certain matrix eigenvalue problem.
Abstract: A recent paper by Donald Gignac describes some difficulties with a certain matrix eigenvalue problem. This paper shows that the difficulties result from the naturue of the matrix itself. Quantities called condition numbers, which measure the sensitivity of the eigenvalues, are computed, and numerical results given.