


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: PerronFrobenius 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 NonHermitian Matrices (Russell Carden)



Lecture 29: 
Eigenvalue perturbation theory for Jordan blocks; Gerschgorin's Theorem
 Moro, Burke, Overton. On the
LidskiiVishikLyusternik 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, BauerFike 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
CayleyHamilton Theorem, characteristic and minimial polynomials



SelfStudy: 
Tensors: basic properties, factorizations, applications
 T. G. Kolda and B. W. Bader,
"Tensor Decompositions and Applications", SIAM Review 51 (2009) 455500.



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), 391407.
 M. W. Berry, S. T. Dumais, G. W. O'Brien,
"Using Linear Algebra for Intelligent Information Retrieval",SIAM Review 37 (1995), 573595.
 A. Björck and G. H. Golub,
"Numerical Methods for Computing Angles Between Linear Subspaces", Math. Comp. 27 (1973), 579594.



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, lowrank 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) = A_{0}+t E.
 avoid_ex2.m: 2d avoidance demo for A(t) nonlinear.
 J. B. Keller,
"Multiple eigenvalues", Linear Algebra Appl. 429 (2008) 22092220.
 Application: photonic band gaps
 Daniel Kressner on spectral band gap computations



Lecture 12: 
CourantFischer 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 (18381922)



Lecture 8: 
Block diagonalization via Sylvester Equations



Lecture 7: 
Sylvester equations
 James Joseph Sylvester (18141897)
 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 108122
and
derivations (1734), pages 162173.
(These are entire volumes of the journal  take a peek at the other articles!)
 Three modes of motion for a 3bead 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, 67:30pm, 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

