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Subsection A.1 Householder 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.