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Conjugate Points, Triangular Matrices, and Riccati Equations

Zeev Nehari
Transactions of the American Mathematical Society
Vol. 199 (Nov., 1974), pp. 181-198
DOI: 10.2307/1996881
Stable URL: http://www.jstor.org/stable/1996881
Page Count: 18
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Conjugate Points, Triangular Matrices, and Riccati Equations
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Abstract

Let A be a real continuous n × n matrix on an interval Γ, and let the n-vector x be a solution of the differential equation x' = Ax on Γ. If [ α, β ] ∈ Γ, β is called a conjugate point of α if the equation has a nontrivial solution vector x = (x1, ..., xn) such that x1(α) = ... = xk(α) = xk+1(β) = ... = xn(β) = 0 for some k ∈ [ 1, n - 1 ]. It is shown that the absence on (t1, t2) of a point conjugate to t1 with respect to the equation x' = Ax is equivalent to the existence on (t1, t2) of a continuous matrix solution L of the nonlinear differential equation L' = [ LA* L-1]τ 0L with the initial condition L(t1) = I, where [ B ]τ 0 denotes the matrix obtained from the n × 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.

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