Showing papers on "Spectrum of a matrix published in 1975"
TL;DR: In this article, the necessary and sufficient conditions for the characteristic roots of a matrix A to lie inside the unit circle are presented, and the following conditions are proved: 1) Linear combinations b i of the coefficients of coefficients of λ √ n of A.
Abstract: In this short paper several theorems related to the necessary and sufficient conditions for the characteristic roots (eigenvalues) of a matrix A to lie inside the unit circle are presented In particular, the following conditions are proved 1) Linear combinations b i of the coefficients of λiin
21 citations
TL;DR: In this paper, it was shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all the eigen values of a real matrix with order 4n lie in the left haft plane.
Abstract: It is shown that the test to determine whether all eigenvalues of a complex matrix of order n lie in a certain sector can be replaced by an equivalent test to find whether all eigenvalues of a real matrix of order 4n lie in the left haft plane.
18 citations
17 citations
TL;DR: In this paper, a way of avoiding certain difficulties in MacFarlane's work on eigenvalues of a rational transfer function matrix G(s) is suggested, and evidence is given for and against the existence of a general stability theorem in terms of these eigen values.
Abstract: A way of avoiding certain difficulties in MacFarlane's work on eigenvalues of a rational transfer function matrix G(s), is suggested. Evidence is then given for and against the existence of a general stability theorem in terms of these eigenvalues. The evidence suggests that no such theorem can be obtained without some condition on G(s). and any general condition on G(s) is unlikely to be simple
11 citations
TL;DR: It is shown that if complete spectral data are provided, the potential function in a Sturm-Liouville operator is uniquely determined almost everywhere and if an operator and its spectrum are given and the potential is presumably known, then the potential corresponding to the second operator can be explicitly found by solving a set of nonlinear ordinary differential equations.
Abstract: It is known that if complete spectral data are provided, the potential function in a Sturm-Liouville operator is uniquely determined almost everywhere. If two such operators have spectra that differ in a finite number of eigenvalues, then the corresponding potential functions will no longer be the same. However, as is demonstrated when the nonidentical eigenvalues are almost equal, then the corresponding potential functions will also be nearly equal almost everywhere. Furthermore, if an operator and its spectrum are given and the potential is presumably known and if a second operator is defined in such a way such that its eigenvalues agree with the eigenvalues of the first operator except for a finite set, then the potential corresponding to the second operator can be explicitly found by solving a set of nonlinear ordinary differential equations. Lastly, it is shown that the procedures discussed here and the Gelfand-Levitan procedures are significantly different, and in fact that the Gelfand-Levitan procedure is almost certainly not well posed.
9 citations
TL;DR: A general multivariable stability theorem in terms of the eigenvalues of G(s), if one exists, should prove to be of considerable use as discussed by the authors, and two related results are presented in this paper.
Abstract: A general multivariable stability theorem in terms of the eigenvalues of G(s), if one exists, should prove to be of considerable use. This paper presents two related results : one involving bounds on the eigenvalues and the other, a special result for two-variable systems.
4 citations
TL;DR: In this article, it was shown how the well-known Schwarz's result about the location of eigenvalues of a matrix in left-half plane can be derived by using an inertia theorem due to Chen.
Abstract: It is shown how the well-known Schwarz's result about the location of eigenvalues of a matrix in left-half plane can be derived by using an inertia theorem due to Chen.
4 citations
TL;DR: In this article, a connection between the number of negative eigenvalues, the parameters of the equation and some properties of the domain was established through the correspondence of the eigen values with those of a related Stekloff problem.
Abstract: This paper deals with several eigenvalue problems that have a finite number of negative eigenvalues. A connection is established between the number of negative eigenvalues, the parameters of the equation and some properties of the domain, through the correspondence of the eigenvalues with those of a related Stekloff problem.
3 citations
TL;DR: In this paper, the authors developed closed-form expressions for f(A) where A is any constant n \times n matrix and f(\lambda) is any analytic function in a region containing the eigenvalues λ i of A, i = 1,2,..., n.
Abstract: In this technical note, we develop closed-form expressions for f(A) where A is any constant n \times n matrix and f(\lambda) is any analytic function in a region containing the eigenvalues λ i of A, i = 1,2, ..., n . The matrix A may have nondistinct eigenvalues.
1 citations
TL;DR: In this article, a best possible upper bound on the product of the eigenvalues of the closed-loop system is obtained, and the relation between the weighting factors of a quadratic performance criterion and the closed loop system is investigated.
Abstract: Explicit relations between the weighting factors of a quadratic performance criterion and the closed-loop system are developed. A best possible upper bound on the product of the eigenvalues of the closed-loop system is attained.
TL;DR: In this article, the authors show that the computation of the transient response of linear time invariant systems can be reduced to the computing of an expanded state transition matrix, by adopting the method of auxiliary states to generate system inputs.
Abstract: In this paper the computation of the transient response of linear time invariant systems is shown to reduce to the computation of an expanded state transition matrix, by adopting the method of auxiliary states to generate system inputs. The evaluation of the composite state transition matrix is shown to be essentially given by the inverse of the transposed confluent Vandermonde matrix of the composite systems' eigenvalues. The inverse of the Vandermonde matrix is given by a now and highly efficient numerical algorithm of the eigenvalue-diagonal transformation type. This algorithm is shown to generate the inversion by a set of recursion equations developed from the composite systems' characteristic equations. These recursion equations are amenable to hand or machine computation and may also be used to evaluate the characteristic equation and its derivatives at sample points other that at its eigenvalues.