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The starting point of analysis of state space models is investigating existence, uniqueness and solution properties such as the semigroup property, and various formulas for the solutions. Several concepts such as the state transition matrix, the matrix exponential, the variations of constants formula (the Cauchy formula), the Peano-Baker series, and the Picard iteration are used to characterize solutions. In this note, a tutorial treatment is given where all of these concepts are shown to be various manifestations of a single abstract method, namely solving equations using an operator Neumann series involving the Volterra operator of forward integration. The matrix exponential, the Peano-Baker series, the Picard iteration, and the Cauchy formula can be "discovered" naturally from this Neumann series. The convergence of the series and iterations is a consequence of the key property of asymptotic nilpotence of the Volterra operator. This property is an asymptotic version of the nilpotence property of a strictly-lower-triangular matrix. 

Analytic perturbation theory for matrices and operators is an immensely useful mathematical technique. Most elementary introductions to this method have their background in the physics literature, and quantum mechanics in particular. In this note, we give an introduction to this method that is independent of any physics notions, and relies purely on concepts from linear algebra. An additional feature of this presentation is that matrix notation and methods are used throughout. In particular, we formulate the equations for each term of the analytic expansions of eigenvalues and eigenvectors as {\em matrix equations}, namely Sylvester equations in particular. Solvability conditions and explicit expressions for solutions of such matrix equations are given, and expressions for each term in the analytic expansions are given in terms of those solutions. This unified treatment simplifies somewhat the complex notation that is commonly seen in the literature, and in particular, provides relatively compact expressions for the non-Hermitian and degenerate cases, as well as for higher order terms.