MATH 5524 is a graduate survey of applicable topics in matrix analysis. Students are expected to arrive with a foundation in basic linear algebra at the undergraduate level. Topics will include: spectral theory, variational properties of eigenvalues, singluar values, eigenvalue perturbation theory, functions of matrices and dynamical systems, nonnegative matrices and Perron-Frobenius theory.

## Classes: |
Monday/Wednesday/Friday, 12:20-1:10pm, McBryde 328 |

## Instructor: |
Mark Embree, embree@vt.edu, 575 McBryde |

## Office Hours: |
Monday 4-5:30pm, Thursday 1:30-3pm, or by appointment |

## Piazza: |
The course Piazza page provides a forum for class discussions. |

Markov chains and card shuffling

riffle_sim.m:
MATLAB demo of riffle shuffling

riffle.m:
MATLAB code by Jonsson and Trefethen giving the transition matrix for the riffle shuffle

Nonnegative matrices.

The Perron-Frobenium Theorem

Intro to Markov chains

Positive matrices.

If $A>0$, then $\rho(A)\in\sigma(A)$ only has $1\times 1$ Jordan blocks.

If $A>0$, then $\rho(A)\in\sigma(A)$ is a simple eigenvalue.

If $A>0$ and $Ay = \lambda y$ with $y\ge 0$, then $\lambda=\rho(A)$ and $y>0$.

Positive matrices.

If $A>0$ and $A x = \lambda x$ with $|\lambda|=\rho(A)$, then $|x|>0$.

If $A>0$ and $A x = \lambda x$ with $|\lambda|=\rho(A)$, then $\lambda=\rho(A)$.

Intro to Markov Chains

Intro to positive matrices

If $A>0$, then $\rho(A)$ is an eigenvalue

Bounds on $\|f(A)\|$ using eigenvalues and eigenvectors, numerical range, pseudospectra

Pseudospectra and eigenvalue conditioning

Bauer-Fike theorems, eigenvalue condition numbers.

Pseudospectra

Download EigTool from GitHub.

Notes on pseudospectra (work in progress): chapter5.pdf (updated 18 April).

The derivative of an eigenvalue for diagonalizable matrices

The eigenvalues of a Jordan block with a perturbed corner entry

Notes on Gerschgorin's Theorem: chapter5.pdf (updated 18 April).

Properties of the numerical range, $W(A)$

Johnson's algorithm for approximating the boundary of $W(A)$

The numerical range (field of values) of matrix and its connection to $\|e^{tA}\|$

**To do:** hw5.pdf, Problem Set 5 is due on Tuesday, 18 April (5pm).

Transient growth in linear dynamical systems: cancellation effects

Asymptotic stability of solutions to $x'(t) = A x(t)$

Potential for transient growth and sensitivity of the eigenvalues of $A$

Functions of matrices: properties of the exponential of a matrix

Functions of matrices: the exponential of a matrix

Functions of matrices: three definitions

Some course notes: chapter4.pdf (updated 17 April).

Singular value potpourri:

- Schatten $p$ norms: $\|A\|_{S-p} = (s_1^p + \cdots + s_r^p)^{1/p}$

- Singular value inequalities, e.g. $s_{j+k-1}(A+B) \le s_j(A)+s_k(B)$.

**To do:** hw4.pdf, Problem Set 4 is due on Tuesday, 4 April (5pm).

Using the SVD to find the minimal norm solutions to
$Ax=b$ and $\min_x \|Ax-b\|$

The Moore-Penrose pseudoinverse

See Section 3.2 of the course notes: chapter3.pdf (updated 1 April).

(Empirical) Principal Component Analysis

Optimal low-rank approximations from the SVD

Introduction to Principal Component Analysis

Some course notes: chapter2.pdf (updated 1 April).

Some course notes: chapter3.pdf (updated 1 April).

Three flavors of the SVD:

- the dyadic decomposition $A = \sum_{j=1}^r s_j^{} u_j^{ } v_j^*$;

- the skinny SVD $A = U\Sigma V^*$ with $\Sigma$ square;

- the full SVD $A = U\Sigma V^*$ with $U$ and $V$ both unitary.

The polar decomposition of a square matrix: $A = H Q$ where $H=U\Sigma U^*$ is positive semidefinite and $Q=UV^*$ is unitary.

Introduction to the singular value decomposition (SVD)

The matrix norm equals the largest singular value: $\|A\| = s_1$.

Postive semi-definite matrices have unique $k$th roots: $A^{1/k} = \sum_{j=1}^n +(\lambda_j)^{1/k} u_j^{} u_j^*$.

Congruent matrices, Sylvester's Law of Inertia, spectrum slicing

No class on March 1 and 3 (instructor traveling). Make-up lectures will be offered after Spring Break.

**To do:** project.pdf. Start thinking about your class project; select by 23 March.

Section 2.4: Two examples illustrating the scarcity of double eigenvalues

Eigenvalues of Jacobi matrices

Eigenvalues of parameterized Hermitian systems

avoid_ex0.m,
avoid_ex1.m,
avoid_ex2.m:
MATLAB demos showing eigenvalue avoidance

Section 2.3: Hermitian matrices

Courant--Fischer characterization of eigenvalues of Hermitian matrices

Some course notes: chapter2.pdf (updated 1 April).

Section 2.2: Cauchy Interlacing Theorem for Hermitian matrices

Some course notes: chapter2.pdf (updated 1 April).

Section 2.1:
Variational characterization of eigenvalues of Hermitian matrices.

Solutions to Problem Set 2: sol2.pdf

**To do:** hwp1.pdf, Pledged Problem Set 1 is due on Friday, 24 February (5pm).

Section 1.8: The Jordan Canonical Form

Jordan form clean-up: algebraic and geometric multiplicity; why we never compute the Jordan form

jordex1.m, jordex2.m: Try computing the Jordan form of these matrices...

Golub and Wilkinson, Ill-Conditioned Eigensystems and the Computation of the Jordan Canonical Form, 1976.

Section 1.8: The Jordan Canonical Form

Derivation of the Jordan form: second half of proof ($\rho \ne 0$)

Fletcher and Sorensen, An Algorithmic Derivation of the Jordan Canonical Form, 1983.

Some course notes: chapter1.pdf (updated 29 March).

Section 1.8: The Jordan Canonical Form

Derivation of the Jordan form: first half of proof ($\rho = 0$)

Fletcher and Sorensen, An Algorithmic Derivation of the Jordan Canonical Form, 1983.

Some course notes: chapter1.pdf (updated 29 March).

Section 1.8: The Jordan Canonical Form

Spectral projectors and nilpotents: $P_j = V_j \widehat{V}_j^*$ and $D_j = V_j R_j \widehat{V}_j^*$

Spectral representation $A = \sum \lambda_j P_j + D_j$.

Section 1.8: The Jordan Canonical Form

Block diagonalization: $A = V {\rm diag}(T_1, \ldots, T_p) V^{-1}$.

Fletcher and Sorensen, An Algorithmic Derivation of the Jordan Canonical Form, 1983.

Solutions to Problem Set 1: sol1.pdf

**To do:** hw2.pdf, Problem Set 2 is due on Tuesday, 14 February (5pm).

Section 1.8: The Jordan Canonical Form

Theorem: The Sylvester equation $AX-XB=C$ has a unique solution if and only if $\sigma(A)\cap\sigma(B) = \emptyset$.

Proof: The Bartels-Stewart algorithm.

Section 1.7: Damped systems in mechanics

shm.m, shmeig.m: MATLAB demos for the damped pendulum

Section 1.4: Reduction to triangular form (Schur decomposition)

Section 1.5: Spectral Theorem for Hermitian matrices

schur_proof.m: MATLAB implementation of the proof of the Schur decomposition

Wikipedia page on Issai Schur (1875-1841)

Some course notes: chapter1.pdf (updated 29 March).

Section 1.3: The resolvent $R(z) := (zI-A)^{-1}$ and the existence of eigenvalues

Wikipedia page on Liouville's Theorem

Some course notes: chapter1.pdf (updated 29 March).

Section 1.3: Proof of the Neumann series for inverting $I-E$ when $\|E\|<1$

**To do:** hw1.pdf, Problem Set 1 is due on Wednesday, 1 February (5pm).

Note: Thursday office hours have moved to 1:30-3pm.

Some course notes: chapter1.pdf (updated 29 March).

Section 1.1: Special matrices: Hermitian, unitary, subunitary, projectors

Section 1.2: Eigenvalues in mechanics

pendulum_demo.m: MATLAB demo of the $n$ modes of an $n$ mass pendulum.

Some course notes: chapter1.pdf (updated 29 March).

Section 1.1: Vector norm, Cauchy-Schwarz, Triangle inequality, induced matrix norm

A good demo: See Cleve Moler's blog post about "eigshow"

Review of the course contract and discussion of book options.

Some course notes: chapter1.pdf (comments/corrections welcome).

Section 1.1: Notation and preliminaries

Section 1.2: Eigenvalues and eigenvectors [today: overview of diagonalization]

Posted 27 February 2017. Due 6 May 2017. (Declare your project by 23 March 2017)

project.pdf: speficiation and grading rubric

Posted 25 April 2017. Due 3 May 2017.

hwp2.pdf: assignment

solp2.pdf: solutions

sept11.m: MATLAB routine for Problem 4

Posted 11 April 2017. Due 18 April 2017. (Late work due 19 April 2017.)

hw5.pdf: assignment

sol5.pdf: solutions

pop.m: MATLAB routine for Problem 4

Posted 28 March 2017. Due 4 April 2017. (Late work due 5 April 2017.)

hw4.pdf: assignment

sol4.pdf: solutions

Posted 15 March 2017. Due 1 April 2017.

hw3.pdf: assignment

cow.mat, planck.mat: MATLAB data files for Problem 4

(cow_A0.csv,
cow_B0.csv,
planck_A0.csv,
planck_B0.csv:
same data, but in .csv format for use with other systems)

Posted 17 February 2017. Due 24 Feburary 2017.

hwp1.pdf: assignment

solp1.pdf: solutions

Posted 6 February 2017. Due 14 Feburary 2017.

hw2.pdf: assignment

sol2.pdf: solutions

Posted 25 January 2017. Due 1 Feburary 2017.

hw1.pdf: assignment

sol1.pdf: solutions

Download a copy of the electronic version of the course contract and tentative schedule.

Final course grades will be thus allocated:

50%: standard problem sets (collaboration encouraged)

35%: pledged problem sets (no collaboration permitted)

15%: end-of-semester project

Virginia Tech's Honor Code applies to all work in this course. Students must uphold the highest ethical standards, abiding by our Honor Code: "As a Hokie, I will conduct myself with honor and integrity at all times. I will not lie, cheat, or steal, nor will I accept the actions of those who do."
From the Office for Undergraduate Academic Integrity:
"Students enrolled in this course are responsible for abiding by the Honor Code. A student who has doubts about how the Honor Code applies to any assignment is responsible for obtaining specific guidance from the course instructor before submitting the assignment for evaluation. Ignorance of the rules does not exclude any member of the University community from the requirements and expectations of the Honor Code. For additional information about the Honor Code, please visit:
www.honorsystem.vt.edu."

While we will not closely follow any single textbook, students are encouraged
to obtain one of the following books, each of which covers most of the topics
we will cover in the lectures.

- Roger A. Horn and Charles R. Johnson,
*Matrix Analysis*, 2nd ed., Cambridge University Press, 2012.

Virginia Tech students have online access to this text.

*This comprehensive reference book is well-suited for those intending to pursue research in matrix theory and related fields.*

- Carl Meyer,
*Matrix Analysis and Applied Linear Algebra*, SIAM, 2001.

Available via Virginia Tech library (2 hour reserve): QA188 .M495 2000

*This textbook is oriented toward advanced undergraduates/beginning graduate students. Those who need a refresher on basic linear algebra concepts will find this a more approachable text.*

You might enjoy dipping in to a few of these supplmental titles

- Rajendra Bhatia, Matrix Analysis, Springer, 1997.

Virginia Tech students have online access to this text.

*This book gives particularly strong coverage to eigenvalue majorization and classical eigenvalue perturbation theory.*

- Harry Dym,
*Linear Algebra in Action*, 2nd ed., AMS, 2013.

Available via Virginia Tech library (2 hour reserve): QA184 .D96 2014

*This book makes particularly good use of complex analysis as a fundamental tool for matrix analysis.*

- Roger A. Horn and Charles R. Johnson,
*Topics in Matrix Analysis*, Cambridge University Press, 1991.

Virginia Tech students have online access to this text.

*This companion to their**Matrix Analyis*text provides a detailed treatment of the field of values, Sylvester and Lyapunov equations, and functions of matrices, among other topics.

- Peter Lancaster and Miron Tismenetsky,
*Theory of Matrices, with Applications**, 2nd ed., Academic Press, 1985.*

*Classic text on advanced matrix theory, particularly strong on canonical forms and matrix polynomials.*

*Peter Lax,**Linear Algebra and Its Applications**, Wiley, 2007.*

*Strong on matrix calculus, avoidance of eigenvalue crossings, abstract normed vector spaces.*

Any student with special needs or circumstances requiring accommodation in this course is encouraged to contact the instructor during the first week of class, as well as the Dean of Students. We will ensure that these needs are appropriately addressed.