This paper proposes a method for estimating eigenvalues and eigenvectors based on a computable residual. We provide guaranteed upper and lower bounds for eigenvalues and establish first-order and second-order error estimates. Additionally, error bounds for eigenvectors are provided, ensuring precise estimates for both eigenvalues and eigenvectors. Numerical experiments have validated the results. Furthermore, the interval algorithm is used for the calculation of residual function, and the resulting eigenvalue bounds are expected to be mathematically accurate.