On characteristical polinomial and eigenvectors in terms of tree-like structure of the graph

While considering the square matrix as an adjacency matrix of a weighted digraph we construct an extended digraph, whose laplacian contains the original matrix as a submatrix. This construction allows us to use the known results on laplacians to study arbitrary square matrices. An eigenvector calculation in parametrical form demonstrates a connection between its components and a tree-like structure of the digraph.