Abstract
Many complex mathematical problems can be formulated as eigenvalue problems. In Exercises 3.1 and 3.2 examples are given where direct computation of eigenvalues and eigenspaces can be carried out. In Exercise 3.3 we show how the eigenvalues of a matrix and its inverse are related, while the eigenvalues of a positive definite matrix are considered in Exercise 3.4. In Exercise 3.5 we see how limits of powers of matrices can be computed via eigenvalues and eigenvectors.
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Notes
- 1.
Compute \(\det (\boldsymbol{M} - t\boldsymbol{I})\).
- 2.
Use \(\text{trace}(\boldsymbol{M}).\)
- 3.
Find the dimension of the eigenspace.
- 4.
Consider the i-th row in \(\boldsymbol{J}\boldsymbol{u} =\lambda \boldsymbol{u}\).
- 5.
Recall the trigonometric formulas sin2A = 2sinAcosA and \(\sin (A + B) +\sin (A - B) = 2\sin A\cos B\).
- 6.
Sum the geometric series.
- 7.
\(\boldsymbol{s}_{j}^{T}\boldsymbol{s}_{j} =\sum _{ k=1}^{m}\sin ^{2}(kj\pi h) = \frac{1} {2}\sum _{k=0}^{m}\big(1 -\cos (2kj\pi h)\big)\). Then use 6.
- 8.
\(1 - x^{2}/2 <\cos x < 1 - x^{2}/2 + x^{4}/24\) for \(x \in (0,\pi )\).
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Lyche, T., Merrien, JL. (2014). Matrices, Eigenvalues and Eigenvectors. In: Exercises in Computational Mathematics with MATLAB. Problem Books in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43511-3_3
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