• This web page describes an activity within the Department of Mathematics at Ohio University, but is not an official university web page.
• If you have difficulty accessing these materials due to visual impairment, please email me at mohlenka@ohio.edu; an alternative format may be available.

# Numerical Analysis: Linear Algebra

## Schedule

### Week 2 (January 21)

#### January 21

Martin Luther King Jr. day holiday, no class

#### 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:
• 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 15 (April 29)

#### Wednesday, May 1

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

Martin J. Mohlenkamp