- jthartwig.net
- Office 470
- Email jth@iastate.edu
- Office hours: MWF 4:10 – 5:00pm, and by appointment

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

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

- [HK] Hoffman and Kunze, Linear Algebra
- [HJ] Horn and Johnson, Matrix Analysis
- [A] Axler, Linear Algebra Done Right (available as a free e-book)
- [H] Hartwig, Linear Algebra
- Other typed notes may be posted here, if needed

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: 55%
- Exam 1: 15% Solutions to Exam 1
- Exam 2: 15% Solutions to Exam 2
- Final: 15%

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.

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

A short summary of each lecture will be posted here.

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

- Fields [HK §1.1], vector spaces [HK §2.1], linear independence [HK §2.3]
- Bases and dimension [HK §2.3], subspaces and sums of subspaces [HK §2.2]
- Matrices, transpose and hermitian adjoint
- Linear maps, coordinates of vectors, matrices of linear transformations, [HK §2.4 and §3.1–§3.4]
- Change of basis [HK §3.4], (External) direct sums of any vector spaces
- Quotient spaces [HK Appendix A.4]
- Tensor products (Lecture notes)
- Modules over \(F[x]\) (Notes on polynomials)
- Minimal polynomial Lecture notes, [HK §6.3], [HJ §3.3]
- Primary Decomposition Theorem, [HK §6.8]
- Abstract Jordan decomposition, [HK §6.8, Thm. 13], Jordan normal form (special case), [HK §7.3]
- Normal Form for Nilpotent Linear Maps and Jordan’s Normal Form, see [H §5 in v0.2], [HK §7.3]
- Evaluating polynomials, Real Jordan Form, [HJ §3.4.1], determinants, [H §5.3-4, §6 in v0.2]
- Invariant Factors I (Lecture notes), [HK §7.4]
- Invariant Factors II (Lecture notes), [HK §7.4]
- (Review)
- (Review)
- Inner products, norms, Cauchy-Schwarz and triangle inequalities [HJ §5.1]; unitary matrices [HJ §2.1]
- More on unitary matrices; \(U(n)\) is a compact topological group [HJ §2.1]
- 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]
- [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]
- Spectral theorem for real normal matrices; Corollary about normal form for real symmetric, skew-symmetric, and orthogonal matrices [HJ §2.5]
- QR factorization and the QR algorithm [HJ §2.6]
- LPU factorization [HJ §3.5] or [H]
- More on norms on vector spaces [HJ §5.2-§5.4]
- Equivalence of norms, dual norms [HJ §5.4] Analytic properties of vector norms
- Matrix norms [HJ §5.6]
- Spectral radius [HJ §5.6]
- Matrix series [HJ §5.6]
- (Review)
- Rayleigh-Ritz [HJ §4.2]
- Applications; Courant-Fischer [HJ §4.2]
- Applications of Courant-Fischer: Weyl’s Theorem, Monotonicity Thm, Interlacing of Eigenvalues
- Sketch of proof of Interlacing; and Aside on Poisson algebras
- [HJ §6.1] Gershgorin Disks
- [HJ §6.2] Gershgorin II: Property (SC) for matrices and strongly connected directed graphs
- [HJ §6.2] Gershgorin III: Return of the disks (A refinement of Gershgorin’s Theorem that applies to matrices satisfying Property (SC).)
- [HJ §7.1-§7.2] Positive Definite Matrices
- [HJ §7.3] Polar Decompisition
- [HJ §7.3] Singular Value Decomposition
- [HJ §7.4] Some applications
- [HJ §7.5] Schur Product Theorem lecture notes
- (Review)
- (Review)

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