[IUCr Home Page] [Commission Home Page]
next up previous contents index
Next: Rules for matrix calculations Up: Matrices and determinants Previous: Motivation

The matrix formalism

Definition (D 2.3.3) A rectangular array of real numbers in $m$ rows and $n$ columns is called a real ($m\times n$) matrix A:

\( \mbox{\textit{\textbf{A}}} = \left( \begin{array}{cccc} A_{11} & A_{12} &
\l...
...&
\ddots & \vdots \\
A_{m1} & A_{m2} & \ldots & A_{mn} \end{array} \right). \)

The left index, running from 1 to $m$, is called the row index, the right index, running from 1 to $n$, is the column index of the matrix. If the elements of the matrix are rational numbers, the matrix is called a rational matrix; if the elements are integers it is called an integer matrix.

Definition (D 2.3.3) An $(n\times n)$ matrix is called a square matrix,

an $(m\times1)$ matrix a column matrix or just a column, and

a $(1\times n)$ matrix a row matrix or, for short, a row.

The index `1' for column and row matrices is often omitted.

Definition (D 2.3.3) Let A be an $(m\times n)$ matrix. The $(n\times m)$ matrix which is obtained from A = ($A_{ik}$) by exchanging rows and columns, i.e. the matrix ($A_{ki}$), is called the transposed matrix A $^{\mbox{\footnotesize {T}}}$.

Example. If \(\mbox{\textit{\textbf{A}}}\hspace{-0.1em} =\hspace{-0.1em} \left(\hspace{-0.1e...
...y}{ccc} 1 & 0 & \bar{1} \\
2 & 4 & \bar{3} \end{array}\hspace{-0.1em}\right)\), then \(\mbox{\textit{\textbf{A}}}^{\mbox{\footnotesize {T}}}\hspace{-0.1em}=\hspace{-...
...{rr} 1 & 2 \\ 0 & 4 \\ \bar{1} & \bar{3}
\end{array} \hspace{-0.1em}\right) \).

(Crystallographers frequently write negative numbers $-z$ as $\bar{z}$, e.g. for MILLER indices or elements of matrices).

Remark. In crystallography point coordinates or vector coefficients are written as columns. In order to distinguish columns from rows (the MILLER indices, e.g., are written as rows), rows are regarded as transposed columns and are thus marked by (..) $^{\mbox{\footnotesize {T}}}$.

General matrices, including square matrices, will be designated by boldface-italics upper case letters A, B, W, ...;

columns by boldface-italics lower case letters a, b, ..., and

rows by (a) $^{\mbox{\footnotesize {T}}}$, (b) $^{\mbox{\footnotesize {T}}}$, ..., see also p. [*], List of symbols.

A square matrix A is called symmetric if A $^{\mbox{\footnotesize {T}}}$ = A, i.e. if $A_{ik}=A_{ki}$ holds for any pair $i,k$.

A symmetric matrix is called a diagonal matrix if $A_{ik}=0$ for $i\ne k$.

A diagonal matrix with all elements $A_{ii}=1$ is called the unit matrix I.

A matrix consisting of zeroes only, i.e. $A_{ik}=0$ for any pair $i,k$ is called the O-matrix.

We shall need only the special combinations $m,n=3,3$ `square matrix'; $m,n=3,1$ `column matrix' or `column' , and $m,n=1,3$ `row matrix' or `row'. However, the formalism does not depend on the sizes of $m$ and $n$. Therefore, and because of other applications, formulae are displayed for general $m$ and $n$. For example, in the Least-Squares procedures of X-ray crystal-structure determination huge ($m\times n$) matrices are handled.


next up previous contents index
Next: Rules for matrix calculations Up: Matrices and determinants Previous: Motivation

Copyright © 2002 International Union of Crystallography

IUCr Webmaster