Showing papers on "Spectrum of a matrix published in 1971"
TL;DR: Dyson's method is adopted here for the so called Gaussian ensembles and confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensemble, the circular and the Gaussian ones.
Abstract: Dyson's method is adopted here for the so called Gaussian ensembles. Incidently this confirms the long cherished belief that the statistical properties of a small number of eigenvalues is the same for the two kinds of ensembles, the circular and the Gaussian ones.
84 citations
TL;DR: In this paper, an upper limit for the number of intersections is derived in terms of the rank of the Gramian of the symmetrized products of order 0, 1, …, n − 1 of A and B.
6 citations
TL;DR: In this article, a modal control theory is developed whereby the loop-gains of a single-input system may be readily calculated using a simple formula for the case when the system matrix has a number of sets of confluent eigenvalues.
Abstract: A modal control theory is developed whereby the loop-gains of a single-input system may be readily calculated using a simple formula for the case when the system matrix has a number of sets of confluent eigenvalues. The simple results derived in this paper are made possible by using the properties of the mode-controllability matrix of the system.
5 citations
4 citations
TL;DR: A method is described to assign arbitrary eigenvalues for linear multivariable systems when the system is not completely controllable, as many arbitrary Eigenvalues as the rank of the controllability matrix are assigned.
Abstract: A method is described to assign arbitrary eigenvalues for linear multivariable systems. When the system is not completely controllable, as many arbitrary eigenvalues as the rank of the controllability matrix are assigned.
2 citations
01 Aug 1971
TL;DR: In this paper, the authors introduce a parameter to improve conditioning in one such method, and give examples to illustrate its effect, and show that the parameter can improve conditioning of an n×n matrix.
Abstract: Due to problems of ill-conditioning, methods of finding the eigenvalues of an n×n matrix which compute its characteristic polynomial have largely been abandoned This letter introduces a parameter to improve conditioning in one such method, and gives examples to illustrate its effect
1 citations
01 Nov 1971
TL;DR: In this article, a simpler and sometimes sharper version of Ruhe's inequality is presented. But the inequality is not applicable to the case where the matrix's eigenvalues are not all distinct.
Abstract: : Recently Axel Ruhe published an inequality from which he could infer that whenever at least one of a matrix's eigenvalues is very ill-conditioned then that matrix must lie very near another whose eigenvalues are not all distinct. The paper exhibits a simpler and sometimes sharper version of that inequality. (Author)
1 citations
TL;DR: In this article, the eigenvalues of two types of band matrices are discussed and sufficient conditions for all of them to be real and simple are derived, with powers of eigen values of a certain band matrix of the first type.
Abstract: This paper discusses the eigenvalues of two types of band matrices.In the first type, the only nonzero elements occur either immediately below the main diagonal or $
u - 1$ places above it (where $
u \geqq 2$). If all the elements in these positions are positive, then such a matrix, of order n, has $[ {{n /
u }} ]$ distinct positive eigenvalues: to prove this, we consider changes of sign of the characteristic polynomial.The second type of matrix discussed differs from a Toeplitz matrix in the first row only; moreover, the $(i,j)$th element is zero if $j \leqq - 2$ or $j \geqq
u $. We derive sufficient conditions for all the eigenvalues to be real and simple: in the proof, we show that these eigenvalues are with powers of eigenvalues of a certain band matrix of the first type.