In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.
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Figure 1. (a) Positive eigenvalues $ \lambda_i({\bf{G}}(\alpha)) $ (green curves, $ i = 1,\ldots,5 $), their best lower bounds derived from the first inequalities in Eqs. (1a) and (1b) in Proposition 1 (blue curves) with $ K = 50 $, and their best upper bounds derived from the same inequalities, still with $ K = 50 $ (red curves); (b) for $ K = 1 $, $ i = 1 $, and each $ \alpha \in [0,1] $: $ \sin(\theta_{1,{\rm min}}(\alpha)) $ (green curve), and smallest upper bound on it, based on the second to last inequalities in Eqs. (11a) and (11b) in Proposition 2 (blue curve)
Figure 2. Beam lattice metamaterials with viscoelastic resonators and their reference periodic cell [19]
Figure 3. Floquet-Bloch spectrum maximizing a low-frequency band gap of a mechanical metamaterial filter: (a) $ 3 $-dimensional representation; (b) projection of the spectrum onto a vertical plane
Figure 4. Floquet-Bloch spectrum maximizing a high-frequency pass band of a mechanical metamaterial filter: (a) $ 3 $-dimensional representation; (b) projection of the spectrum onto a vertical plane
Figure 5. Floquet-Bloch spectrum maximizing a trade-off between a low-frequency bang gap and a high-frequency pass band of a mechanical metamaterial filter: (a) $ 3 $-dimensional representation; (b) projection of the spectrum onto a vertical plane
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Figures(5)
(a) Positive eigenvalues
Beam lattice metamaterials with viscoelastic resonators and their reference periodic cell [19]
Floquet-Bloch spectrum maximizing a low-frequency band gap of a mechanical metamaterial filter: (a)
Floquet-Bloch spectrum maximizing a high-frequency pass band of a mechanical metamaterial filter: (a)
Floquet-Bloch spectrum maximizing a trade-off between a low-frequency bang gap and a high-frequency pass band of a mechanical metamaterial filter: (a)