MATH 6640-100 (11486), Spring 2019

Numerical Analysis: Linear Algebra

Syllabus

Schedule

Week 1 (January 14)

Week 2 (January 21)

January 21

Martin Luther King Jr. day holiday, no class

Tuesday January 22, 9am: homework week1 due

January 23

Information:
  • Today we will work on some topics that you identified in the paper.
  • In \(\LaTeX\) either $...$ or \(...\) can be used to indicate inline math. The form \(...\) is better within html using MathJax, but markdown works better with $...$. Similarly, markdown prefers $$...$$ over \[...\] for displayed math.
Homework:
  • Create a Jupyter notebook week2.ipynb. Put your name in a markdown cell. Do your work this week in it.
  • Read about Norm and Matrix norm. Using these or other sources, answer the following, each in a markdown cell. Remember to cite your sources.
    • State the defining properties of a norm.
    • Let \(f(\mathbf{x})\) be the function defined on vectors \(\mathbf{x}=(x_1,x_2,x_3)\in \mathbb{R}^3\) by \[f(\mathbf{x})=|x_1|+2|x_2|+3|x_3|\,.\] Determine whether or not \(f\) is a norm.
    • Define what it means for a matrix norm to be induced by a vector norm.
    • Show that if \(A\) and \(B\) are square matrices and the matrix norm is induced by a vector norm, then \(\|AB\|\le \|A\|\, \|B\|\).
  • State the defining properties of a seminorm.
  • State the defining properties of a quasinorm.

January 25

Information:
  • Today we will work on some topics that you identified in the paper.
  • We will also go through an existence proof for the SVD.
Homework:
  • Read about the Singular Value Decomposition (SVD).
    • State its definition.
    • State the formula for the matrix norm induced by \(\|\cdot\|_2\) in terms of the singular values.
    • State the formula for the Frobenius norm in terms of the singular values.
    • State the formula for the Shatten norm in terms of the singular values. What is the Shatten quasinorm? What is the nuclear norm?
    • State the formula for the matrix rank in terms of the singular values. What can you say about the singular values of a rank-1 matrix?

Week 3 (January 28)

January 28

Tuesday January 29, 9am: homework week2 due

January 30

February 1

Week 4 (February 4)

Week 5 (February 11)

Week 6 (February 18)

Week 7 (February 25)

Week 8 (March 4)

Spring Break

Week 9 (March 18)

Week 10 (March 25)

Week 11 (April 1)

Week 12 (April 8)

Week 13 (April 15)

Week 14 (April 22)

Week 15 (April 29)

Wednesday, May 1

Final Exam 12:20-2:20pm. Your final project notebook is due at 2:20pm.


Martin J. Mohlenkamp

Last modified: Wed Jan 23 15:58:25 UTC 2019