Conjugate Points, Triangular Matrices, and Riccati Equations

Let be a real continuous <!-- MATH $n \times n$ --> matrix on an interval , and let the -vector be a solution of the differential equation on . If <!-- MATH $[\alpha ,\beta ] \in \Gamma ,\beta$ --> is called a conjugate point of if the equation has a nontrivial solution vector <!-- MATH $x = ({x_1},{\kern 1pt} \ldots ,{x_n})$ --> such that <!-- MATH ${x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0$ --> for some <!-- MATH $k \in [1,n - 1]$ --> .