On some properties of trigonometric matrices. (English) Zbl 0642.34024

Došlý, Ondřej

Čas. Pěstování Mat. 112, 188-196 (1987).

The pair of \(n\times n\) matrices \(\{\) S(x),C(x)\(\}\) which satisfy \(Y'=Q(x)Z\), \(Z'=-Q(x)Y\) and the initial conditions \(S(a)=0\), \(C(a)=E\), where E is the \(n\times n\) identity matrix and Q(x) is an \(n\times n\) symmetric matrix of continuous functions on [a,\(\infty)\), are called trigonometric matrices. These matrices have been used to study the self- adjoint matrix differential system \((F(x)Y')'+G(x)Y=0\) by means of the generalized Prüfer transformation. In this paper, the author studies oscillation properties of trigonometric matrices and extend some well- known trigonometric formulae to trigonometric matrices.

Reviewer: N.Parhi

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

trigonometric matrices; self-adjoint matrix differential system; Prüfer transformation; oscillation properties