# Spring 2024

### Instructor

• Kean Fallon, Office Hours: 1:00 – 3:00pm on Tuesdays in Carver 410

### Time and Place

• Lectures: MWF 3:20pm – 4:10pm in Carver 232

### Course Content

Advanced topics in linear algebra including canonical forms; unitary, normal, Hermitian and positive-definite matrices; variational characterizations of eigenvalues.

The course is roughly divided into three parts:

Part 1 focuses on algebraic aspects, and we cover [HK, Ch. 1–7].

Part 2 concerns geometric aspects, from [HK, Ch. 8–9] and [HJ, Ch. 2, 4–5].

Part 3 will be on analytic properties, from [HJ, Ch. 4, 6– 7].

Weekly homework, due most Fridays.

### Homework

Homework sets will be posted here. They are due most Fridays. You are strongly encouraged to use LaTeX to typeset your homework solutions. You may download and use the homework LaTeX file as a starting point. Overleaf is a popular online LaTeX environment which offers helpful tutorials.

1. (Due Jan 26) Homework 1, (LaTeX file)
2. (Due Feb 2) Homework 2, (LaTeX file)
3. (Due Feb 9) Homework 3, (LaTeX file)
4. (Due Feb 16) Homework 4, (LaTeX file)
5. (Due March 8) Homework 5, (LaTeX file)
6. (Due March 22) Homework 6, (LaTeX file)
7. (Due March 29) Homework 7, (LaTeX file)
8. (Due April 12) Homework 8, (LaTeX file)
9. (Due April 26) Homework 9, (LaTeX file)
10. (Not due) Practice Problems for Final Exam

### Lecture Summary

A short summary of each lecture will be posted here.

HK = Hoffman and Kunze, HJ = Horn and Johnson (1st Edition), H = Hartwig

1. Fields [HK §1.1], vector spaces [HK §2.1], linear independence [HK §2.3]
2. Bases and dimension [HK §2.3], subspaces and sums of subspaces [HK §2.2]
3. Matrices, transpose and hermitian adjoint
4. Linear maps, coordinates of vectors, matrices of linear transformations, [HK §2.4 and §3.1–§3.4]
5. Change of basis [HK §3.4], (External) direct sums of any vector spaces
6. Quotient spaces [HK Appendix A.4]
7. Tensor products (Lecture notes)
8. Modules over $$F[x]$$ (Notes on polynomials)
9. Minimal polynomial Lecture notes, [HK §6.3], [HJ §3.3]
10. Primary Decomposition Theorem, [HK §6.8]
11. Abstract Jordan decomposition, [HK §6.8, Thm. 13], Jordan normal form (special case), [HK §7.3]
12. Normal Form for Nilpotent Linear Maps and Jordan’s Normal Form, see [H §5 in v0.2], [HK §7.3]
13. Evaluating polynomials, Real Jordan Form, [HJ §3.4.1], determinants, [H §5.3-4, §6 in v0.2]
14. Invariant Factors I (Lecture notes), [HK §7.4]
15. Invariant Factors II (Lecture notes), [HK §7.4]
16. (Review)
17. (Review)
18. Inner products, norms, Cauchy-Schwarz and triangle inequalities [HJ §5.1]; unitary matrices [HJ §2.1]
19. More on unitary matrices; $$U(n)$$ is a compact topological group [HJ §2.1]
20. Matrices $$A$$ such that $$A^\ast\sim A^{-1}$$ [HJ §2.1]; Schur’s Unitary Triangularization Theorem [HJ §2.3]; some consequences [HJ §2.4]
21. [HJ Thm.2.2.2]; “Diagonalizable matrices are dense in the set of all matrices” [HJ Thm.2.4.8]; normal matrices and the Spectral Theorem [HJ §2.5]
22. Spectral theorem for real normal matrices; Corollary about normal form for real symmetric, skew-symmetric, and orthogonal matrices [HJ §2.5]
23. QR factorization and the QR algorithm [HJ §2.6]
24. LPU factorization [HJ §3.5] or [H]
25. More on norms on vector spaces [HJ §5.2-§5.4]
26. Equivalence of norms, dual norms [HJ §5.4] Analytic properties of vector norms
27. Matrix norms [HJ §5.6]
29. Matrix series [HJ §5.6]
30. (Review)
31. Rayleigh-Ritz [HJ §4.2]
32. Applications; Courant-Fischer [HJ §4.2]
33. Applications of Courant-Fischer: Weyl’s Theorem, Monotonicity Thm, Interlacing of Eigenvalues
34. Sketch of proof of Interlacing; and Aside on Poisson algebras
35. [HJ §6.1] Gershgorin Disks
36. [HJ §6.2] Gershgorin II: Property (SC) for matrices and strongly connected directed graphs
37. [HJ §6.2] Gershgorin III: Return of the disks (A refinement of Gershgorin’s Theorem that applies to matrices satisfying Property (SC).)
38. [HJ §7.1-§7.2] Positive Definite Matrices
39. [HJ §7.3] Polar Decompisition
40. [HJ §7.3] Singular Value Decomposition
41. [HJ §7.4] Some applications
42. [HJ §7.5] Schur Product Theorem lecture notes
43. (Review)
44. (Review)

### Statement on Free Expression

Iowa State University supports and upholds the First Amendment protection of freedom of speech and the principle of academic freedom in order to foster a learning environment where open inquiry and the vigorous debate of a diversity of ideas are encouraged. Students will not be penalized for the content or viewpoints of their speech as long as student expression in a class context is germane to the subject matter of the class and conveyed in an appropriate manner.