It is only possible to perfectly diagonalize certain systems of linear differential equations. For the more general cases, it is possible to "block-diagonalize" the system into what is known as Jordan Canonical Form. This video explores these various options and derives the fully general Jordan form.
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0:00 A tale of two "A" matrices
1:47 When it's possible to diagonalize a matrix with eigenvectors
6:15 Computing eigenvectors and generalized eigenvectors
20:03 Case of complex conjugate eigenvalues
23:10 Case of repeated eigenvalues
25:35 3x3 degenerate matrix
29:37 Jordan canonical form for general matrix