The chapter introduces the perturbation theory for a linear operator \( X \) in an \( n \)-dimensional complex vector space \( R \). The operator \( X \) is expressed as a perturbation series: \[ X = X_0 + \varepsilon X_1 + \varepsilon^2 X_2 + \cdots \] where \( \varepsilon \) is a small parameter, and the matrices \( X_i \) are defined in a fixed basis \( e_1, \ldots, e_n \) and do not depend on \( \varepsilon \). The perturbation theory aims to find the asymptotic expansions of the eigenvalues and eigenvectors of \( X \) as \( \varepsilon \to 0 \). For distinct eigenvalues \( \lambda_1, \ldots, \lambda_n \) of the leading operator \( X_0 \), the eigenvalues and eigenvectors of \( X \) are expanded as: \[ \lambda_j^* = \lambda_j + \varepsilon \lambda_j^{(1)} + \varepsilon^2 \lambda_j^{(2)} + \cdots, \quad e_j^* = e_j + \varepsilon e_j^{(1)} + \varepsilon^2 e_j^{(2)} + \cdots \] When some eigenvalues coincide, the expansions become more complex, involving fractional powers of \( \varepsilon \). The goal is to find an operator \( C(\varepsilon) \) such that the matrix of the operator \( M = C^{-1} X C \) is in Jordan form. The operator \( C \) is assumed to be of the form: \[ C = E + \varepsilon C_1 + \varepsilon^2 C_2 + \cdots \] where \( E \) is the identity operator and the matrices \( C_i \) do not depend on \( \varepsilon \). The chapter provides formulas and definitions for the perturbation theory, including the expansion of \( C \) in terms of a series involving the operator \( S \): \[ C = e^S = E + S + \frac{1}{2!} S^2 + \cdots \] where \( S = \varepsilon S_1 + \varepsilon^2 S_2 + \cdots \) and the matrices of the operators \( S_j \) do not depend on \( \varepsilon \).The chapter introduces the perturbation theory for a linear operator \( X \) in an \( n \)-dimensional complex vector space \( R \). The operator \( X \) is expressed as a perturbation series: \[ X = X_0 + \varepsilon X_1 + \varepsilon^2 X_2 + \cdots \] where \( \varepsilon \) is a small parameter, and the matrices \( X_i \) are defined in a fixed basis \( e_1, \ldots, e_n \) and do not depend on \( \varepsilon \). The perturbation theory aims to find the asymptotic expansions of the eigenvalues and eigenvectors of \( X \) as \( \varepsilon \to 0 \). For distinct eigenvalues \( \lambda_1, \ldots, \lambda_n \) of the leading operator \( X_0 \), the eigenvalues and eigenvectors of \( X \) are expanded as: \[ \lambda_j^* = \lambda_j + \varepsilon \lambda_j^{(1)} + \varepsilon^2 \lambda_j^{(2)} + \cdots, \quad e_j^* = e_j + \varepsilon e_j^{(1)} + \varepsilon^2 e_j^{(2)} + \cdots \] When some eigenvalues coincide, the expansions become more complex, involving fractional powers of \( \varepsilon \). The goal is to find an operator \( C(\varepsilon) \) such that the matrix of the operator \( M = C^{-1} X C \) is in Jordan form. The operator \( C \) is assumed to be of the form: \[ C = E + \varepsilon C_1 + \varepsilon^2 C_2 + \cdots \] where \( E \) is the identity operator and the matrices \( C_i \) do not depend on \( \varepsilon \). The chapter provides formulas and definitions for the perturbation theory, including the expansion of \( C \) in terms of a series involving the operator \( S \): \[ C = e^S = E + S + \frac{1}{2!} S^2 + \cdots \] where \( S = \varepsilon S_1 + \varepsilon^2 S_2 + \cdots \) and the matrices of the operators \( S_j \) do not depend on \( \varepsilon \).