The eigenvalues of reducible matrices over min-plus algebra

The eigenvalue is an essential concept contained in the square matrix. In max-plus algebra, a square matrix is depicted as a precedence graph. Precedence graphs may be defined in two forms, namely strongly connected graphs and not strongly connected graphs. An irreducible matrix represents graph that is strongly connected, whereas the reducible matrix represents graphs that is not strongly connected. The aim of this study is to formulate the methods in finding the eigenvalues of reducible matrices over min-plus algebra. There is an isomorphic relationship between min-plus and max-plus algebra. Hence, the eigenvalues of reducible matrices over min-plus algebra are able to be evaluated according to a concept from eigenvalues of reducible matrices over max-plus algebra. The outcome of this study is methods in determining the eigenvalues of reducible matrices over min-plus algebra. We obtain the determination of eigenvalues by transforming the reducible matrices into Frobenius Normal Form (FNF), making the elements in the principal diagonal irreducible square submatrices. Furthermore, we determine the eigenvalue of each submatrix. The eigenvalue of a spectral submatrix is the eigenvalues of the matrix.