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→Methods of solution: modified linearization notation to better match the reference |
→Systems of differential equations: Clarify alpha parameter and construction of X, Lambda |
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:<math> M q''(t) +C q'(t) + K q(t) = 0 </math>
Where <math> q(t) \in \mathbb{R}^n </math>, and <math> M, C, K \in \mathbb{R}^{n\times n}</math>. If all quadratic eigenvalues of <math> Q(\lambda) = \lambda^2 M + \lambda C + K </math> are distinct, then the solution can be written in terms of the quadratic eigenvalues and right quadratic eigenvectors as
:<math>
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, 2n} </math> are the quadratic eigenvalues, <math> X = [x_1, \ldots, x_{2n}] \in \mathbb{R}^{n, 2n} </math> are the <math> 2n</math> right quadratic eigenvectors, and <math> \alpha = [\alpha_1, \cdots, \alpha_{2n}]^\top \in \mathbb{R}^{2n}</math>
[[Stability theory]] for linear === Finite element methods ===
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