Throughout many scientific and engineering fields, including control theory, quantum mechanics, advanced dynamics, and network theory, a great many important applications rely on the spectral decomposition of matrices. Traditional methods such as the power iteration method, Jacobi eigenvalue method, and QR decomposition are commonly used to compute the eigenvalues and eigenvectors of a square and symmetric matrix. However, these methods suffer from certain drawbacks: in particular, the power iteration method can only find the leading eigen-pair (i.e., the largest eigenvalue and its corresponding eigenvector), while the Jacobi and QR decomposition methods face significant performance limitations when facing with large scale matrices. Typically, even producing approximate eigenpairs of a general square matrix requires at least O(N3) time complexity, where N is the number of rows of the matrix. In this work, we exploit the newly developed memristor technology to propose a low-complexity, scalable memristorbased method for deriving a set of dominant eigenvalues and eigenvectors for real symmetric non-negative matrices. The time complexity for our proposed algorithm is O(N2/Δ) (where Δ governs the accuracy). We present experimental studies to simulate the memristor-supporting algorithm, with results demonstrating that the average error for our method is within 4%, while its performance is up to 1.78X better than traditional methods.