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In this video, we take a closer look at the Jordan normal form, focusing in particular on how to calculate it. The Jordan normal form is the closest approximation to a diagonal matrix for cases where matrices are not fully diagonalizable on their own. This transformation is especially useful when computing the matrix exponential, since a straightforward solution exists in the Jordan form. Such calculations are required, for example, when deriving a discrete state-space model, as I demonstrated in last month’s video ( • Discrete Statespace representation: Why, H... ). The matrix chosen for this example requires the computation of generalized eigenvectors. It was selected specifically to illustrate an additional step: For this matrix, the eigenvectors cannot be used directly. Instead, a linear combination of both eigenvectors must be employed. The reason behind this requirement is explained, and the demonstration shows why this is the case. Videos mentioned: For more background, see last month’s video on discrete system models and the matrix exponential (mentioned around 00:25): • Discrete Statespace representation: Why, H... Tags/ Keywords: Jordan Normal Form Tutorial, Matrix Exponential Explained, Discrete State-Space Models, Eigenvalues and Eigenvectors, Generalized Eigenvectors (Hauptvectors), Linear Algebra Applications, Jordan Normal Form Step-by-Step Calculation Chapters: 00:00 - Introduction 01:23 - Recipe for Calculation 03:19 - Eigenvectors & Hauptvectors 09:26 - Building T and J 11:47 - Matrix Exponential 14:23 - Larger Matrices 15:55 - Outlook