Special matrices

GUIDING QUESTION: How can I exploit matrix structure in solving linear systems?

Motivation

In many cases we can exploit structure in the coefficent matrix to develop faster and more stable numerical methods, and to guarantee the convergence of our methods.

We'll consider a few examples in this module.
An $n\times n$ matrix $A$ is strictly diagonally dominant (SDD) if for each row, the diagonal entry is strictly larger than the sum of the other elements in the row (in absolute value).
That is, $A$ is SDD if $$|a_{\color{var(--emphColor)}{ii}}| > \sum_{\color{var(--emphColor)}{j}\neq i} |a_{i\color{var(--emphColor)}{j}}|$$ for each $i = 1, \ldots, n$.
The matrix $A = \begin{bmatrix} 3 & -1 & 1\\ 2 & -6 & 3\\ -9 & 7 & -20 \end{bmatrix}$
is SDD \begin{align} \\ \\ \\ \end{align}
The matrix $A = \begin{bmatrix} \bbox[3pt, border: 3pt solid var(--emphColor)]{3} & \color{var(--emphColor)}{-1} & \color{var(--emphColor)}{1}\\ 2 & -6 & 3\\ -9 & 7 & -20 \end{bmatrix}$
is SDD because \begin{align} |\color{var(--emphColor)}{3}| &> |\color{var(--emphColor)}{-1}| + |\color{var(--emphColor)}{1}| = 2,\\ \\ \\ \end{align}
The matrix $A = \begin{bmatrix} 3 & -1 & 1\\ \color{var(--emphColor)}{2} & \bbox[3pt, border: 3pt solid var(--emphColor)]{-6} & \color{var(--emphColor)}{3}\\ -9 & 7 & -20 \end{bmatrix}$
is SDD because \begin{align} |3| &> |-1| + |1| = 2,\\ |\color{var(--emphColor)}{-6}| &> |\color{var(--emphColor)}{2}| + |\color{var(--emphColor)}{3}| = 5, \text{ and}\\ \\ \end{align}
The matrix $A = \begin{bmatrix} 3 & -1 & 1\\ 2 & -6 & 3\\ \color{var(--emphColor)}{-9} & \color{var(--emphColor)}{7} & \bbox[3pt, border: 3pt solid var(--emphColor)]{-20} \end{bmatrix}$
is SDD because \begin{align} |3| &> |-1| + |1| = 2,\\ |-6| &> |2| + |3| = 5, \text{ and}\\ |\color{var(--emphColor)}{-20}| &> |\color{var(--emphColor)}{-9}| + |\color{var(--emphColor)}{7}|= 16. \end{align}