This paper proposes an analog circuit approach to computing the eigenvectors corresponding to all the eigenvalues of a class of positive definite matrices. The proposed analog computational model can be considered specialized analog computers relying on strongly simplified models of elements. The key features of the proposed analog computational model are asynchronous parallel processing, continuous-time dynamics and high-speed computational capability. We show analytically and by simulations that the proposed circuit can provide the desired eigenvectors with arbitrarily small error during an elapsed time of only a few characteristic time constants of the circuit. In addition, the parameters of the circuit can be obtained from the given matrix without any computations. For the wider use, we also generalize this proposed method for the case in which the matrix takes the complex values. As a result, this proposed approach is satisfactory for many real-time applications fields.