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Propositions from Spectral applications

All propositions from Spectral applications.

[PROPOSITION]

If

D=\operatorname{diag}(\lambda_1,\ldots,\lambda_m),

then, for every non-negative integer $k$ ,

D^k=\operatorname{diag}(\lambda_1^k,\ldots,\lambda_m^k).

The matrix stays diagonal because each coordinate direction is acted on independently.

Part of the thread Roots, modes and evolution, in 7 places.

A linear evolution splits into modes $r^n$ or $e^{rt}$ , and the roots $r$ decide growth, decay and oscillation.