Scalars

Scalars are denoted by lower case Greek letters, e.g., $\alpha, \beta, \gamma, \ldots$ . We will assume that these scalars come from a given set, $S$ , where $S$ may be the set of integers ( $I$ ), real numbers ( $R$ ), or complex numbers integers ( $C$ ).

Vectors

Vectors: are denoted by lower case Roman letters, e.g., $a, b, c, \ldots$ .

A (column vector of length $m$ consists of an m-tuple of coordinates (elements or entries) arranged as a column,

$x = \left( \begin{array}{c} \chi_0 \\ \chi_1 \\ \vdots \\ \chi_{m-1} \end{array} \right) $ or $x = \left( \begin{array}{c} \chi_0 \\ \hline \chi_1 \\ \hline \vdots \\ \hline \chi_{m-1} \end{array} \right).$

A row vector of length $n$ is thought of as a column vector of length $n$ that has been transposed:

$x^T = \left( \begin{array}{c c c c} \chi_0 & \chi_1 & \cdots & \chi_{n-1} \end{array} \right) $ or $x^T = \left( \begin{array}{c | c | c | c} \chi_0 & \chi_1 & \cdots & \chi_{n-1} \end{array} \right).$

(In both cases, the lines are meant for emphasis.)

Matrices

Matices: are denoted by upper case Roman letters, e.g., $A, B, C, \ldots$ .

Indices, dimensions

Lower case Roman letters $i, j, k, m, n, p$ are often used for integer scalars.

Partitioning matrices and vectors

We will often work with partitioned vectors or matrices:

$\begin{tabular}{| l I p{4in} |} \hline $x \rightarrow \FlaTwoByOne { x_T } { x_B }$ & Partition $ x $ into two subvectors, $ x_T $ (read: $ x $-Top) and $ x_B $ (read: $ x $-Bottom). \\ \whline $x \rightarrow \FlaTwoByOne { \chi_T } { x_B }$ & Identify the first element of $ x $. (Note the use of $ \chi $ to indicate the first element is a scalar.) \\ \hline $A \rightarrow \FlaTwoByOne { A_T } { A_B }$ & Partition $ A $ horizontally into two submatrices, $ A_T $ (read: $ A $-Top) and $ A_B $ (read: $ A $-Bottom) \\ \hline $A \rightarrow \FlaTwoByOne { a_T^T } { A_B }$ & Identify the first row of $ A $. (Note that the lower case letter $ a $ together with the superscript"$\ ^T$" indicates that $ a_T^T $ is a row vector.) \\ \whline $x^T \rightarrow \FlaOneByTwo { x_L^T } { x_R^T }$ & Partition row vector $ x^T $ into two row vectors, $ x_L^T $ (read: $ x $-Left) and $ x_R^T $ (read: $ x $-Right). \\ \hline $x^T \rightarrow \FlaOneByTwo { x_L^T } { \chi_R }$ & Identify the last element of row vector $ x $. \\ \hline $A \rightarrow \FlaOneByTwo { A_L }{ A_R }$ & Partition $ A $ vertically into two submatrices, $ A_L $ (read: $ A $-Left) and $ A_R $ (read: $ A $-Right) \\ \hline $A \rightarrow \FlaOneByTwo { A_L } { a_R }$ & Identify the last column of $ A $. \\ \whline $A \rightarrow \FlaTwoByTwo { A_{TL} }{ A_{TR} } { A_{BL} }{ A_{BR} }$ & Partition $ A $ into four quadrants. $ T $, $ B $, $ L $, and $ R $ stand for Top, Bottom, Left, Right, respectively. \\ \hline $A \rightarrow \FlaTwoByTwo { \alpha_{TL} }{ a_{TR}^T } { a_{BL} }{ A_{BR} }$ & Identify the Top-Left element ($ \alpha_{TL} $), rest of the first row ($ a_{TR}^T $), rest of the first column ($ a_{BL} $), and the remainder ($A_{BR}$) of $ A $. Notice that the fact that $ \alpha_{TL} $ is a scalar dictates that $ a_{TR}^T $ has one row and is therefore a row vector. It also dictates that $ a_{BL} $ has one column and is therefore a (column) vector. \\ \hline \end{tabular}$

Repartitioning vectors and matrices

$\begin{tabular}{| l I l |} \hline Repartitioning & Symantic meaning \\ \whline $ \FlaTwoByOne { x_T } { x_B } \rightarrow \FlaThreeByOneB { x_0 } { \chi_1 } { x_2 } $ & $ \begin{array}{l} { x_T \rightarrow x_0 } \\ { x_B \rightarrow \FlaTwoByOneSingleLine { \chi_1 }{ x_2 } } \end{array} $ \\ \hline $ \FlaTwoByOne { A_T } { A_B } \rightarrow \FlaThreeByOneT { A_0 } { A_1 } { A_2 } $ & $ \begin{array}{l} { A_T \rightarrow \FlaTwoByOneSingleLine { A_0 }{ A_1 } }\\ { A_B \rightarrow A_2 } \end{array} $ \\ \hline $ \FlaOneByTwo{ x_L^T }{ x_R^T } \rightarrow \FlaOneByThreeR{ x_0^T }{ x_1^T }{ x_2^T } $ & $ \begin{array}{l} x_L^T \rightarrow x_0^T \\ x_R^T \rightarrow \FlaOneByTwoSingleLine{ x_1^T }{ x_2^T } \end{array} $ \\ \hline $ \FlaOneByTwo{ A_L }{ A_R } \rightarrow \FlaOneByThreeL{ A_0 }{ A_1 }{ A_2 } $ & $ \begin{array}{l} A_L \rightarrow \FlaOneByTwoSingleLine{ A_0 }{ A_1 } \\ A_R \rightarrow A_2 \end{array} $ \\ \hline $ \FlaTwoByTwo { A_{TL} }{ A_{TR} } { A_{BL} }{ A_{BR} } \rightarrow \FlaThreeByThreeBR { A_{00} }{ a_{01} }{ A_{02} } { a_{10}^T }{ \alpha_{11} }{ a_{12}^T } { A_{20} }{ a_{21} }{ A_{22} } $ & $ \begin{array}{l l} A_{TL} \rightarrow A_{00} & A_{TR} \rightarrow \FlaOneByTwoSingleLine{ a_{01} }{ A_{02} } \\ A_{BL} \rightarrow \FlaTwoByOneSingleLine{ a_{10}^T }{ A_{20} } & A_{BR} \rightarrow \FlaTwoByTwoSingleLine { \alpha_{11} }{ a_{12}^T } { a_{21} }{ A_{22} } \end{array} $ \\ \hline $ \FlaTwoByTwo { A_{TL} }{ A_{TR} } { A_{BL} }{ A_{BR} } \rightarrow \FlaThreeByThreeTL { A_{00} }{ A_{01} }{ A_{02} } { A_{10} }{ A_{11} }{ A_{12} } { A_{20} }{ a_{21} }{ A_{22} } $ & $ \begin{array}{l l} A_{TL} \rightarrow \FlaTwoByTwoSingleLine { A_{00} }{ A_{01} } { A_{10} }{ A_{11} } & A_{TR} \rightarrow \FlaTwoByOneSingleLine{ A_{02} }{ A_{12} } \\ A_{BL} \rightarrow \FlaOneByTwoSingleLine{ A_{20} }{ A_{21} } & A_{BR} \rightarrow A_{22} \end{array} $ \\ \hline $ \FlaTwoByOne { x_T } { x_B } \leftarrow \FlaThreeByOneB { x_0 } { \chi_1 } { x_2 } $ & $ \begin{array}{l} { x_T \leftarrow x_0 } \\ { x_B \leftarrow \FlaTwoByOneSingleLine { \chi_1 }{ x_2 } } \end{array} $ \\ \hline $ \FlaTwoByOne { A_T } { A_B } \leftarrow \FlaThreeByOneT { A_0 } { A_1 } { A_2 } $ & $ \begin{array}{l} { A_T \leftarrow \FlaTwoByOneSingleLine { A_0 }{ A_1 } }\\ { A_B \leftarrow A_2 } \end{array} $ \\ \hline $ \FlaOneByTwo{ x_L^T }{ x_R^T } \leftarrow \FlaOneByThreeR{ x_0^T }{ x_1^T }{ x_2^T } $ & $ \begin{array}{l} x_L^T \leftarrow x_0^T \\ x_R^T \leftarrow \FlaOneByTwoSingleLine{ x_1^T }{ x_2^T } \end{array} $ \\ \hline $ \FlaOneByTwo{ A_L }{ A_R } \leftarrow \FlaOneByThreeL{ A_0 }{ A_1 }{ A_2 } $ & $ \begin{array}{l} A_L \leftarrow \FlaOneByTwoSingleLine{ A_0 }{ A_1 } \\ A_R \leftarrow A_2 \end{array} $ \\ \hline \end{tabular}$

LinearAlgebraWiki: Notation (last edited 2006-12-01 15:17:03 by RobertVanDeGeijn)