MATH 510 Linear Algebra

Spring 2024

Instructor

Grader

Time and Place

Texts and Resources

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].

Exams and Grading

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]
  28. Spectral radius [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)

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