Dynamic mode decomposition (DMD) is a powerful tool for analyzing nonlinear systems, originally introduced in fluid mechanics. This paper presents a theoretical framework for DMD as the eigendecomposition of an approximating linear operator, generalizing DMD to a broader class of datasets, including nonsequential time series. The authors demonstrate the utility of this approach by introducing novel sampling strategies that enhance computational efficiency and mitigate noise effects. They also define the concept of linear consistency, which helps explain the pitfalls of applying DMD to rank-deficient datasets. The paper shows that DMD strengthens connections with Koopman operator theory and other methods, including the eigensystem realization algorithm (ERA) and linear inverse modeling (LIM). It is shown that under certain conditions, DMD is equivalent to LIM. The paper also discusses the practical implications of the new DMD framework, including the benefits of applying DMD to nonsequential time series and the importance of linear consistency in ensuring accurate results. The paper provides examples demonstrating the effectiveness of DMD in analyzing both sequential and nonsequential datasets, as well as its ability to capture dynamics in noisy environments. The paper concludes with a discussion of the connections between DMD and other methods, emphasizing the importance of theoretical foundations in understanding and applying DMD to nonlinear systems.Dynamic mode decomposition (DMD) is a powerful tool for analyzing nonlinear systems, originally introduced in fluid mechanics. This paper presents a theoretical framework for DMD as the eigendecomposition of an approximating linear operator, generalizing DMD to a broader class of datasets, including nonsequential time series. The authors demonstrate the utility of this approach by introducing novel sampling strategies that enhance computational efficiency and mitigate noise effects. They also define the concept of linear consistency, which helps explain the pitfalls of applying DMD to rank-deficient datasets. The paper shows that DMD strengthens connections with Koopman operator theory and other methods, including the eigensystem realization algorithm (ERA) and linear inverse modeling (LIM). It is shown that under certain conditions, DMD is equivalent to LIM. The paper also discusses the practical implications of the new DMD framework, including the benefits of applying DMD to nonsequential time series and the importance of linear consistency in ensuring accurate results. The paper provides examples demonstrating the effectiveness of DMD in analyzing both sequential and nonsequential datasets, as well as its ability to capture dynamics in noisy environments. The paper concludes with a discussion of the connections between DMD and other methods, emphasizing the importance of theoretical foundations in understanding and applying DMD to nonlinear systems.