# Jianchao Zhu

## On Linear Differential Equations with Functionally Commutative Coefficient Matrices

Let C and R be the fields of complex and real numbers, respectively. Let Mn(C) be the space of matrices of order n over C. Let I ⊆ R and D ⊆ C. A matrix-valued function FI → Mn(C) is said to be functionally commutative on I if

F(t)F(τ) = F(τ)F(t), for all t, τ in I,

and F is said of the proper if

F(t) = f(tA)

where A in Mn(C) and fI × C → C is a scalar function. The system of linear differential equations

x' = PxPI → Mn(C),

will be called semiproper if P is functionally commutative on I. It is known that a semiproper system has a closed form fundamental solution matrix

where the matrix exponential function is defined by the power series

Therefore the problem of solving a semiproper system amounts to that of finding a finite form for the matrix exponential. in this thesis, systematic approaches are developed for finding finite form analytical solutions for semiproper systems. Stability criteria for semiproper systems are also presented. To this end, functionally commutative matrix functions are investigated in terms of proper matrix functions.