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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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 \(x \in (0,\pi )\).
References
J. Bastien, Introduction à l’analyse numérique: Applications sous MATLAB, Dunod, 2003.
A. Biran, M. Breiner, MATLAB6 for Engineers, Prentice Hall, 2002, 3th ed.
A. Björck, Numerical Methods for Least Squares Problems, SIAM, 1996.
K. Chen, P. Giblin, A. Irving, Mathematical Explorations with MATLAB, Cambrige Univ. Press, 1999.
E. Cohen, R. F. Riesenfeld, G. Elber, Geometric Modeling with Splines: An Introduction, A.K. Peters, Ltd., 2001.
J. Cooper, MATLAB Companion for Multivariable Calculus (a), Academic Press, 2001.
M. Crouzeix, A. Mignot, Analyse numérique des équations différentielles, Masson, 1989.
C. de Boor, A Practical Guide to Splines, Springer-Verlag, 2001, Rev. Ed.
P. Deuflhard, A. Hohmann, Numerical Analysis in Modern Scientific Computing, An Introduction, Springer-Verlag, 2003, 2nd ed.
A. Fortin, Analyse numérique pour ingénieurs, Ed. de l’Ecole Polytechnique de Montréal, 1995.
W. Gander, J. Hřebíček, Solving Problems in Scientific Computing using MAPPLE and MATLAB, Springer-Verlag, 2004, 4th ed.
S. Godounov, V. Riabenki, Schémas aux différences, MIR, 1977.
G.H. Golub, C.F. Van Loan, Matrix Computations, The Johns Hopkins Univ. Press, 1996, 3rd Ed.
A. Greenbaum, Iterative mMethods for Solving Linear Systems, SIAM, 1997.
F. Gustafsson, N. Bergman, MATLAB for Engineers Explained, Springer-Verlag, 2003.
D.J. Higham, N. Higham, MATLAB Guide, SIAM, 2005, 2nd Ed.
A. Kharab, R. B. Guenther, Introduction to Numerical Methods (a): a MATLAB Approach, Chapman and Hall, 2002.
P. Lascaux and R. Théodor, Analyse numérique matricielle appliquée à l’art de l’ingénieur, vol 1 and 2, Dunod, 2004.
A.J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, 2005.
P. Linz, R. Wang, Exploring Numerical Methods: An Introduction to Scientific Computing using MATLAB, Jones and Barlett Pub., 2003.
D. Marsh, Applied Geometry for Computer Graphics and CAD, Springer-Verlag, 1999.
C. B. Moler, Numerical Computing with MATLAB, SIAM, 2004.
G.M. Phillips, Interpolation and Approximation by Polynomials, CMS Books in Mathematics, Springer-Verlag 2003.
A. Quarteroni, R. Sacco, F. Saleri Numerical Mathematics, Springer-Verlag, 2000.
A. Quarteroni, R. Sacco, F. Saleri Scientific Computing with MATLAB, Springer-Verlag, 2003.
D. Salomon, Curves and Surfaces for Computer Graphics, Springer-Verlag, 2006.
M. Schatzman, Numerical Analysis: A Mathematical Introduction, Oxford Univ. Press, 2002.
B.J. Schroers, Ordinary Differential Equations: a Practical Guide, Cambridge Univ. Press, 2011.
E. Süli, D. Mayers, An introduction to Numerical Analysis, Cambridge Univ. Press, 2003.
L.N. Trefethen & D. Bau III, Numerical Linear Algebra, SIAM, 1997.
C.F. Van Loan, K.-Y. Daysy Fan, Insight Through Computing, SIAM, 2010.
H. B. Wilson, L. H. Turcotte, D. Halpern, Advanced Mathematics and Mechanics Applications using MATLAB, Chapman and Hall, 2003, 3rd ed.
K. Yosida, Functional Analysis, Springer-Verlag, 1980, 6–th ed.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-662-43511-3_3
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-43510-6
Online ISBN: 978-3-662-43511-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)