Quadratic eigenvalue problem: Difference between revisions

Revision as of 22:26, 9 March 2023

Revision as of 22:43, 9 March 2023

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q(t) = \sum_{j=1}^{2n} \alpha_j x_j e^{\lambda_j t} = X e^{\Lambda t} \alpha

</math>

Where <math>\Lambda = \text{Diag}([\lambda_1, \ldots, \lambda_{2n}]) \in \mathbb{R}^{2n, \times 2n} </math> are the quadratic eigenvalues, <math> X = [x_1, \ldots, x_{2n}] \in \mathbb{R}^{n, \times 2n} </math> are the <math> 2n</math> right quadratic eigenvectors, and <math> \alpha = [\alpha_1, \cdots, \alpha_{2n}]^\top \in \mathbb{R}^{2n}</math> is a parameter vector determined from the initial conditions on <math> q</math> and <math> q'</math>.

[[Stability theory]] for linear systems can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.