Eigenvalues and eigenvectors of matrix are useful in both pure and applied mathematics such as differential equations and dynamical systems (Fuhrmann in A Polynomial Approach to Linear Algebra. New York, Springer, 1996). They are also applied to investigate the problems in engineering, physics and chemistry (Lay, et al., in Linear Algebra and Its Applications, Pearson Education, 2016). This chapter studies the characteristic structure of Boolean matrix over the Boolean field \(\mathcal {B}\) . Using the eigenvalues and eigenvectors, the characteristic polynomial, similarity, diagonalization and generalized eigenvectors of Boolean matrix are systematically discussed. Finally, the relation between eigenvectors and attractors of BNs is revealed.