MATH 2371 Matrices and linear operators 2 (2015)
University of Pittsburgh

What's new!
o Jan 8: We covered Rayleigh quotient and proved that eigenvectors are exactly critical points of the Rayleigh quotient (Section 8).
o The first homework is now posted. It is due on Friday Jan. 22.
o We finished Chapter 8. We also proved the Schur factorization theorem (Appendix 10). We have started with Chapter 9. We reviewed derivative of a map between normed vector spaces (from calculus). We stated (generalized) chain rule and Leibniz rule (for matrix valued and vector valued functions). We also stated the formula for the derivative of a multi-linear function. We used it to calculate the derivative of det(A(t)) in terms of trace.
o We covered Chapter 9 to the end of exponential function of matrices.
o Feb. 6: We have covered Chapter 9 up to determinant of positive matrices. We started to prove that det is a log-concave on the space of positive matrices.
o Feb. 6: HW3 is now posted. It is due Friday Feb. 12.
o Feb. 7: As announced earlier, there will be a midterm on Monday Feb. 15 (in class). It covers material we have discussed so far (+ material from last term). That is, spectral theory of self-adjoint maps (Chap 8.), calculus of matrix-valued functions (Chap. 9) and positive matrices (Chap. 10). We skipped a few theorems in Chap. 9 and 10 and you won't need them. I will try to make the problems similar to the prelim.
o Feb. 13: Midterm is this Monday. Make sure you know how to do the homework problems and the topics related to them.
o Feb. 13: Some topics you need to know for midterm : Spectral theory of self-adjoint matrices, Rayleight quotient, (Hilbert-Courant) min-max principle, derivative of matrix valued functions A(t), derivative of product of matrices, derivative of inverse of a matrix, exponential map, A=D+N decomposition (every matrix is sum of a diagonalizable matrix and a nilpotent matrix), derivative of determinant, positive matrices, every positive matrix has square root, Schur decomposition (every matrix can be made upper triangular by conjugating with a unitary matrix, see Appendix 10 in Lax), Gram matrix, every Gram matrix is non-negtive, concave function, determinant is a log-concave function, determinant of a positive matrix is less than product of diagonal entries, Hadamard theorem about determinant, every principal submatrix of a positive matrix is positive, Sylvester'c criterion for positivity.
o Feb. 24: On Wednesday we discussed different matrix decomposition theorems: QR Decomposition (also known as Iwasawa decomposition), Polar Decomposition and Singular Value Decomposition.
o Feb. 26: We started reviewing material in the course. We reviewed determinant and did couple of prelim problems about determinant. We will continue with determinant next time.
o Feb. 26: I posted the solutions to MT.
o Feb 26: I posted HW4.
o March 7: I posted HW5.
o March 19: HW6 is now posted.
o April 20: Final exam is on Wednesday April 27 in class.
o April 20: Prelim exam is on Monday May 2.
o April 25: Prelim topics posted.
o April 25: Extra topics discussed in the course but will not be in the prelim and final: tensor product and brief discussion of multi-linear algebra. Entry-wise positive matrices and Perron’s theorem, some convexity and proof of Birkhoff-Von Neumann theorem on doubly stochastic matrices.
o April 27: Final exam is posted below.


Online material and weekly homework
o Prelims for previous years linear algebra can be found at: Previous prelim exams

o Prof Jiang’s notes from last year. These cover material for two course MATH2370 and MATH2371. We did not cover all the material in these notes. I am posting them here just for extra reference.

o Some practice problems from last term . Some of these are from previous prelim exams, and some other ones are just for practice or review.

o Some practice problems from last years useful to prepare for prelim.

o Homework 1

o Homework 2

o Homework 3

o Solutions to midterm

o Homework 4

o Homework 5

o Homework 6

o Practice test 1

o Practice test 2

o List of topics discussed in the course (for review)

o Final exam

Course Information
Text: The course is based on Lax’s “Linear algebra and applications”.

Course description: Spectral theory of self-adjoint operators (Section 8), calculus of matrix valued functions (Section 9) and matrix inequalities (Section 10), plus few extra topics about determinant and multi-linear algebra.

Last term topics: Linear transformations of finite dimensional vector spaces are studied in a semi-abstract setting. The emphasis is on topics and techniques which can be applied to other areas, e.g. bases and dimension, matrix representation, linear functionals, duality, canonical forms, vector space decomposition, inner products and spectral theory.

Time of lectures: Monday-Wednesday-Friday 10:00AM - 10:50AM
Location of lectures: 158 Benedum

o There will be midterm(s) and weekly homeworks.

o Homework is assigned each Friday and is due the next Friday in class. I will announce when the first homework is posted.

o Announcements are through email (make sure your email address in Peoplesoft is working).
o Final grade is 50% final exam, 25% midterm(s) and 25% homeworks (lowest homework will be dropped).

o Important dates:

Jan. 6: first class
Jan. 18: Martin Luther King’s Day(no class)
March 6 - 13: Spring break (no class)
April 27: Final exam.
April 29: Last class before prelim.
May 2: prelim exam.

Instructor's Information
Email: kavehk AT pitt.edu
Office: Thackeray #424
Office hours: 11:00AM-12:00PM Tuesdays or by appointment. You can drop by I am usually in.
Office phone: 412-624-8331