CAAM 440 · Applied Matrix Analysis

Spring 2012 · Rice University


Notes and Supplemental Material

Lecture 36: Nonnegative matrices in Markov chains
- PageRank
- Card shuffling: Aldous and Diaconis, Bayer and Diaconis, Trefethen and Trefethen
Lecture 35: Nonnegative matrices in the analysis of networks
- Estrada and Higham. Network properties revealed through matrix functions, SIAM Review, 2010.
- More network analysis at Peter Mucha's site
Lecture 34: Nonnegative matrices: Perron-Frobenius Theorem
- Georg Frobenius
Lecture 33: Positive matrices: Perron's Theorem, continued
Lecture 32: Positive matrices: Perron's Theorem
- Oskar Perron
Lecture 31: Nonnegative matrices: introduction
- See Chapter 8 of Meyer, Matrix Analysis and Applied Linear Algebra, 2000.
Lecture 30: Eigenvalue majorization and Ritz values for nonsymmetric matrices
- Marshall, Olkin, Arnold. Inequalities: Theory of Majorization and Its Applications, 2nd ed, 2009
- Ritz Value Localization for Non-Hermitian Matrices (Russell Carden)
Lecture 29: Eigenvalue perturbation theory for Jordan blocks; Gerschgorin's Theorem
- Moro, Burke, Overton. On the Lidskii-Vishik-Lyusternik Perturbation Theory
   for Eigenvalues of Matrices with Arbitrary Jordan Structure
, 1997.
- Gerschgorin's Theorem
Lecture 28: Eigenvalue perturbation theory
- Tosio Kato, Perturbation Theory for Linear Operators, 2nd ed., 1976.
Lecture 27: Pseudospectra, Bauer-Fike Theorem
- EigTool for computing pseudospectra and the numerical range.
Lecture 26: Numerical range and Pseudospectra
- Slides from Elgersburg lectures: 1, 2, 3, 4, 5
- Trefethen and E., Spectra and Pseudospectra, 2005.
Lecture 25: The numerical range (field of values)
- Horn and Johnson, Topics in Matrix Analysis, Chapter 1 (on reserve at Fondren).
- Crouzeix, Numerical Range and Functional Calculus in Hilbert Space, 2007
- Crouzeix, Bounds for Analytical Functions of Matrices, 2004
- Anne Greenbaum slides on Crouzeix's conjecture
Lecture 24: The matrix exponential: "the hump" and transient behavior
- Trefethen et al., Hydrodynamic Stability Without Eigenvalues, 1993.
Lecture 23: The matrix exponential: series definition; commutativity rules
- Moler and Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix..., 2003.
- Higham, The Scaling and Squaring Method for the Matrix Exponential Revisited, 2005.
Lecture 22: Functions of matrices: contour integral definition
Review of Cauchy's Theorem, Cauchy's integral formula, and its derivative form (essentially the residue theorem).
N. J. Higham, Functions of Matrices: Theory and Computation", SIAM, 2008.
Lecture 21: Functions of matrices: introduction
Cayley-Hamilton Theorem, characteristic and minimial polynomials
Self-Study: Tensors: basic properties, factorizations, applications
- T. G. Kolda and B. W. Bader, "Tensor Decompositions and Applications", SIAM Review 51 (2009) 455-500.
Lecture 20: Latent Semantic Indexing; angles between subspaces/subspace intersections
- S. Deerwester, S. T. Dumais, G. W. Furnas, T. K. Landauer, R. Harshman
"Indexing by latent semantic analysis", J. Am. Info. Sci. 41 (1990), 391-407.
M. W. Berry, S. T. Dumais, G. W. O'Brien, "Using Linear Algebra for Intelligent
Information Retrieval",SIAM Review 37 (1995), 573-595.

- A. Björck and G. H. Golub, "Numerical Methods for Computing
Angles Between Linear Subspaces", Math. Comp. 27 (1973), 579-594.
Lecture 19: Principal component analysis
- Notes on PCA
- I. T. Jolliffe, Principal Component Analysis, 2nd ed., 2002 (available online via SpringerLink)
   See especially the examples in Chapter 4.
Lecture 18: Singular value decomposition: polar decomposition, eigenfaces
- Notes on the polar decomposition
- Mean image for our class, and first six leading eigenfaces:
Lecture 17: Singular value decomposition: derivation, low-rank aproximation
- CAAM 453 notes on the SVD: lecture17.pdf, lecture18.pdf
Lecture 16: Uniqueness of the HPD matrix square root; begin SVD
- CAAM 453/553 notes on polynomial interpolation: lecture9.pdf, lecture10.pdf
Lecture 15: Positive definite matrices; simultaneous diagonalization and commutativity of matrix products
- hessdemo.m: Eigenvales and eigenvectors of a Hessian
Lecture 14: Hermitian tridiagonal matrices
- tridwilk.m: Wilkinson's example of a tridiagonal with a *nearly* multiple eigenvalue
Lecture 13: Eigenvalue Avoidance
- avoid_ex1.m: 2d avoidance demo for A(t) = A0+t E.
- avoid_ex2.m: 2d avoidance demo for A(t) nonlinear.
- J. B. Keller, "Multiple eigenvalues", Linear Algebra Appl. 429 (2008) 2209-2220.
- Application: photonic band gaps
- Daniel Kressner on spectral band gap computations
Lecture 12: Courant-Fischer Minimax Theorem, Intro to Eigenvalue Avoidance
- Biography of Richard Courant
Lecture 11: Cauchy's Interlacing Theorem
- interlace.m: MATLAB demo of interlacing
- Cauchy's Cours d'Analyse, 1821. (English translation) See Note III in the appendix.
Lecture 10: Maximum energy generated by a dynamical system: Rayleigh quotients
- Some notes on Hermitian matrices [updated 23 February]
- Nobel biography of Lord Rayleigh
- Rayleigh's Theory of Sound (two volumes, 1877 and 1878)
- Not to be missed: Richard Tapia's "Math at Top Speed" talk tomorrow
- Engineering Grads: Spring Break Entrepreneurship Trek
Lecture 9: Jordan Canonical Form
- Camille Jordan (1838-1922)
Lecture 8: Block diagonalization via Sylvester Equations
Lecture 7: Sylvester equations
- James Joseph Sylvester (1814-1897)
- sylvester.m: MATLAB code to solve Sylvester equations with real matrices (Sorensen/Zhou)
Lecture 6: Left eigenvectors; spectral projectors
Some physically interesting matrices are not diagonalizable
- shm.m: Dynamics of a damped harmonic oscillator
- shm.eigm: Eigenvalues of a harmonic oscillator as a function of damping
Lecture 5: Spectral Theorem for Hermitian matrices
Spectral Theorem for Diagonalizable matrices
Some physically interesting matrices are not diagonalizable
- shm.m: Dynamics of a damped harmonic oscillator
- shm.eigm: Eigenvalues of a harmonic oscillator as a function of damping
Lecture 4: Schur triangular factorization
- schur_proof.m MATLAB implementation of our proof of the Schur factorization
Lecture 3: Resolvents, Neumann expansion, eigenvalues
Every matrix has at least one eigenvalue
- Liouville's Theorem
Lecture 2: Notation; inner products and norms; introduction to eigenvalues
- pendulum_demo.m
- Daniel Bernoulli on the compound pendulum: results (1733), pages 108-122
     and derivations (1734), pages 162-173.
     (These are entire volumes of the journal -- take a peek at the other articles!)
- Three modes of motion for a 3-bead pendulum (Thanks Jeff H. and Jeff B.!)
Lecture 1: Overview; example from network theory
- Notes for early lectures [updated 23 February]
- VIGRE summer undergrad research: learn more at the RMC Grand Hall, 6-7:30pm, Wednesday 11 January

Some supplemental resources:

Gilbert Strang's linear algebra videos from MIT
Rajendra Bhatia, Matrix Analysis, Springer, 1997
Harry Dym, Linear Algebra in Action, AMS, 2007
F. R. Gantmacher, Theory of Matrices, volume 1 and volume 2
Peter Lancaster and Miron Tismenetsky, The Theory of Matrices, 2nd ed., Academic Press, 1985
Peter Lax, Linear Algebra with Applications, Wiley, 1997
Carl Meyer, Matrix Analysis and Applied Linear Algebra, SIAM 2000
L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton, 2005