Time-variant fractional systems have many applications. For example, they can be used for system identiﬁcation of lithium-ion batteries. However, the analytical solution of the time-variant fractional state space equation is missing so far. To overcome this limitation, the present paper introduces a novel matrix approach, namely the generalized Peano-Baker Series, which is comparable to the transition matrix in the case of ordinary systems. Using this matrix, the solution of the time-variant fractional state space equation is derived. The initialization process is taken into account, which has been proven to be a crucial part for fractional operator calculus. Following this initialization, a modiﬁed deﬁnition of a fractional state is presented.