Dimensions Of Solution Spaces Of Two Classes Of Matrix Equations

The theory of Kronecker canonical forms is quite classical,it also plays an important role in the fields of symmetric linear differential equations and matrix equations.For a long time,people have made a lot of researches on the canonical form theory and calculation method of matrix pairs.In modern mathematics,combinatorics and graph theory,as active research fields,are applied to many branches of mathematics,and are widely used in natural science and communication networks.Based on the research results of domestic and overseas mathematicians,in this thesis,we mainly calculate the Kronecker canonical form of the incidence matrix pair of bicyclic directed graphs and the dimensions of the solution spaces of two classes of matrix equations related to Kronecker canonical forms.In this thesis,we first prove that an undirected graph induced by the tensor product of a directed path and a bicyclic directed graph is a tree under the condition that two loops are relatively prime.Furthermore,when the number of vertices of the three kinds of bicyclic directed graphs are the same,invertible matrix pairs are constructed respectively so that their incidence matrix pairs can be transformed into the same Kronecker canonical form,and we give a method to transform the incidence matrix pairs into Kronecker canonical forms.On the basis of previous studies on matrix equations and explicit formulas for the Kronecker invariants of a matrix pencil,we study the relationship between the two classes of related matrix equations for the Kronecker canonical form of the incidence matrix pair of bicyclic digraphs,and the dimensions of their solution spaces are calculated.