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## UnitA.1Householder notation

Alston Householder introduced the convention of labeling matrices with upper case Roman letters ($A \text{,}$ $B \text{,}$ etc.), vectors with lower case Roman letters ($a \text{,}$ $b \text{,}$ etc.), and scalars with lower case Greek letters ($\alpha \text{,}$ $\beta \text{,}$ etc.). When exposing columns or rows of a matrix, the columns of that matrix are usually labeled with the corresponding Roman lower case letter, and the the individual elements of a matrix or vector are usually labeled with "the corresponding Greek lower case letter," which we can capture with the triplets $\{ A, a, \alpha \} \text{,}$ $\{ B, b, \beta \} \text{,}$ etc.

\begin{equation*} A = \left( \begin{array}{c | c | c | c} a_0 \amp a_1 \amp \cdots \amp a_{n-1} \end{array} \right) = \left( \begin{array}{c | c | c | c} \alpha_{0,0} \amp \alpha_{0,1} \amp \cdots \amp \alpha_{0,n-1} \\ \alpha_{1,0} \amp \alpha_{1,1} \amp \cdots \amp \alpha_{1,n-1} \\ \vdots \amp \vdots \amp \amp \vdots \\ \alpha_{m-1,0} \amp \alpha_{m-1,1} \amp \cdots \amp \alpha_{m-1,n-1} \\ \end{array} \right) \end{equation*}

and

\begin{equation*} x = \left( \begin{array}{c} \chi_0 \\ \chi_1 \\ \vdots \\ \chi_{m-1} \end{array} \right), \end{equation*}

where $\alpha$ and $\chi$ is the lower case Greek letters "alpha" and "chi," respectively. You will also notice that in this course we start indexing at zero. We mostly adopt this convention (exceptions include $i \text{,}$ $j \text{,}$ $p \text{,}$ $m \text{,}$ $n \text{,}$ and $k \text{,}$ which usually denote integer scalars. We summarize our notational conventions in Appendix A.