Conjugate Points, Triangular Matrices, and Riccati Equations

Zeev Nehari
1974 Transactions of the American Mathematical Society  
Let A be a real continuous n x n matrix on an interval T, and let the n-vector x be a solution of the differential equation x = Ax on T. If [a, ß) ET, ß is called a conjugate point of a if the equation has a nontrivial solution vector x = (xj.xn) such that x,(a) = ... = xk(a) = xk+l(ß) = -= xn(ß) = 0 for some k e [ 1, n -1 ]. It is shown that the absence on (ij, t2) of a point conjugate to fj with respect to the equation x' = Ax is equivalent to the existence on (f., f,) of a continuous matrix
more » ... continuous matrix solution L of the nonlinear differential equation L' = [LA*L~ ]TrL with the initial condition L(t¡) = I, where [B]t0 denotes the matrix obtained from the n X n matrix B by replacing the elements on and above the main diagonal by zeros. This nonlinear equation-which may be regarded as a generalization of the Riccati equation, to which it reduces for n = 2-can be used to derive criteria for the presence or absence of conjugate points on a given interval. Let A = Ait) be a continuous real-valued n x n matrix on a real interval T. We consider the differential equation
doi:10.2307/1996881 fatcat:pxpmnf3m5vc63lib6szc3x2pxy