Showing papers on "Spectrum of a matrix published in 1980"

Eigenvalue assignment in linear periodic discrete-time systems†

Abstract: This paper considers the stabilization of the linear periodic discrete-time system through the use of linear periodic state-variable feedback. Let the transition matrix of the closed-loop system be Y(t, s). Then the stability of the closed-loop system depends on eigenvalues of Y(τ,0), where τ is period. It is shown that complete reachability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of the eigenvalues to Y(τ, 0) by choosing a suitable state feedback.

A new concept of eigenvalues and eigenvectors and its applications

Abstract: A new concept of "eigenvalues" and "eigenvectors" of A(t) is introduced. This new concept reduces to the conventional one as a special case when A(t) has constant "eigenvectors." Its applications to algebraic transformation and diagonal realization are given.

Scaling variables and interpretation of eigenvalues in principal component analysis of geologic data

Abstract: The dominant feature distinguishing one method of principal components analysis from another is the manner in which the original data are transformed prior to the other computations. The only other distinguishing feature of any importance is whether the eigenvectors of the inner product-moment of the transformed data matrix are taken directly as the Q-mode scores or scaled by the square roots of their associated eigenvalues and called the R-mode loadings. If the eigenvectors are extracted from the product-moment correlation matrix, the variables, in effect, were transformed by column standardization (zero means and unit variances), and the sum of the p-largest eigenvalues divided by the sum of all the eigenvalues indicates the degree to which a model containing pcomponents will account for the total variance in the original data. However, if the data were transformed in any manner other than column standardization, the eigenvalues cannot be used in this manner, but can only be used to determine the degree to which the model will account for the transformed data. Regardless of the type of principal components analysis that is performed—even whether it is Ror Q-mode—the goodness-of-fit of the model to the original data is given better by the eigenvalues of the correlation matrix than by those of the matrix that was actually factored.

On the Lyapunov matrix equation

Abstract: In this paper the inequality which is satisfied by the determinant of the solution of the Lyapunov matrix equation A'Q + QA = - D is presented. The result makes possible a lower estimate of product eigenvalues of the matrix Q and dependence from eigenvalues of the matrices A and D . This result corresponds to those presented in [2] and [3], where an estimate of the extremal eigenvalues of the matrix Q is presented. This estimate depends on the eigenvalues of the matrices A and D .

Shift‐operator techniques for the classification of multipole‐phonon states. II. Eigenvalues of the quadrupole shift operator O0l

Abstract: By the aid of previously derived relations involving shift operators and their products, a set of rules is set up for the determination of eigenvalues of the scalar Hermitian shift operator O0l in the space of R(5) nuclear quadrupole‐phonon states. The eigenvalues are listed for all seniority states with v<8.

On a shift of eigenvalues of a matrix by using a solution of an algebraic Riccati equation

Abstract: This paper is concerned with a property of a certain transformation of a matrix in terms of eigenvalues. The transformation, which results in a circle-symmetric shifting of eigenvalues, is derived by using a solution of an algebraic Riccati equation. A concept of the circle-symmetry is introduced.

A Note on the Asymptotic Expansion of Eigenvalues

Abstract: Under certain conditions k-term asymptotic expansions of the eigenvalues of a matrix can be deduced from a k-term asymptotic expansion of the matrix.

Concerning the Eigenvalues of $DA$ under Variations of the Entries of D and A

Abstract: Let A be a positive definite matrix and D a nonsingular, nonnegative diagonal matrix. This paper is a study of the eigenvalues of $DA$ under variations between zero and infinity of specified entries of D and A. Intervals containing these eigenvalues are determined and these intervals are seen to be the best possible.

Eigenvalues of symmetric matrices: System theory conditions for distinctness

Abstract: We derive the interlacing properties of the eigenvalues of a symmetric matrix with those of its largest principal submatrix using root-locus ideas. We then give a sufficient condition for distinctness of eigenvalues based on the minimality of an associated linear system.