An analytical function f(A) of an arbitrary nxn constant matrix A is determined and expressed by the ''fundamental formula'', the linear combination of constituent matrices. The constituent matrices Z k h , which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z k h of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z k h can be found by the linear combination of matrices, that may be further simplified by writing them in ''super column matrix'' forms. Finally, a typical example is provided to show the merit of several approaches for the constituent matrices of a given matrix A.