5. Spectral decomposition#
Introduction. What’s spectral (or eigenvalue) decomposition? Arbitrarieness. When is it possible? Special matrices. Properties and theorems (Cayley-Hamilton)
\[\mathbf{A} \mathbf{u}_i = s_i \mathbf{u}_i\]Generalized spectral decompositions. Generalized spectral decomposition of first order systems; generalized spectral deocmposition of second order systems1
\[\mathbf{A} \mathbf{u}_i = s_i \mathbf{B} \mathbf{u}_i\]\[\left[ s_i^2 \mathbf{M} + s_i \mathbf{C} + \mathbf{K} \right] \mathbf{u}_i = \mathbf{0} \ .\]Sensitivity of spectral decomposition. …
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It’s likely that mainly underdamped systems with simultaneously diagonalizable matrices will be treated, as this kind of systems often arises in structural dynamics of elastic structures with small structural damping.