Eigenvalue Theorem Research Articles

A generalization of GF method to crystal vibrations

A hermitian dynamical matrix in terms of Cartesian displacement coordinates is derived starting from a harmonic force constant matrix expressed in terms of the Wilson's internal displacement coordinates. The method accounts both for inter- and intramolecular coupling. The frequency spectrum can be obtained from the dynamical matrix without directly solving the secular equation by the application of Dean's negative eigenvalues theorem.

Vibrational Analysis of Random Polymers

The frequency distributions of some isotopically random polymers are calculated by applying Dean's negative eigenvalue theorem. It is shown that the results of calculation are in good agreement with the infrared spectra of polymers of cis-CHD = CHD and trans-CHD = CHD, giving strong confirmation to the previous view that the polymerization by a Ziegler-type catalyst occurs with the cis opening of the double bond. The spectra of copolymers of CH2 = CH2, CHD = CD2, and CD2 = CD2 are also discussed. The effects of randomness and deuterium content on the frequency distribution are studied.

An eigenvalue theorem for nonlinear second-order differential equations

An eigenvalue theorem for nonlinear second-order differential equations

Notes on the use of matrix theory in the analysis of linear feedback circuits

A set of simultaneous first-order differential equations describing the behaviour of a simple class of linear electrical networks is derived and expressed as a partitioned matrix equation. Some basic relationships of linear feedback theory are then derived in matrix terms. These are: the interpretation of transfer functions in terms of the system characteristic matrix, the relationship between transfer functions in the presence and absence of feedback, and the extension of root locus techniques to the study of the loci of the eigenvalues of the system matrix. Attention is then drawn to a set of martix eigenvalue theorems which become available when feedback problems are formulated in this way, and an example is given of their use.

A comparison theorem for eigenvalues of normal matrice

A comparison theorem for eigenvalues of normal matrice