A novel approach to determination of a transition function with repeated eigenvalues constant matrix

An analytical function of a matrix with repeated eigenvalues is expressed in terms of constituent matrices. Two approaches to computing the constituent matrices are then presented. For a special case of a companion matrix, the computation can be greatly simplified. 1.FORMULATION For a given nxn constant matrix A, eigenvalues are determined by solving the characteristic polynomial of degree n, det(sI-A)= () k r k m k n p p n p s s a λ − = Π ∑ = = − 1 0 ∑ = = m k k n r 1 (1) That is, the matrix A has m eigenvalues 1 λ , 2 λ , …, m λ with multiplicities 1 γ , 2 γ , …, m γ , respectively. Let f(s) be an analytical function of s, than any analytical matrix function of A , f(A) may be written as the expression of the n-term sum [1]: f(A)= , !) (1 1 0) (kh m k h k h k h f Ζ ∑ ∑ = − = γ λ (2) k γ are called constituent matrices [2]. The main objective of this article is to derive the most efficient technique to compute kh Ζ from a given constant matrix A. In general the computation is very much involved, especially for large value of n with high multiplicity. In a usual approach [2], the evaluation of kh Ζ may be made by successively