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##
The matrix formalism

**Definition** (D 2.3.3) A rectangular array of real numbers in rows and
columns is called a *real* () *matrix*
**A**:

The left index, running from 1 to , is called the *row index*,
the right index, running from 1 to , 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 matrix is called a *square matrix*,

an matrix a *column matrix* or just
a *column*, and

a 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 matrix. The
matrix which is obtained from **A** = () by exchanging rows and
columns, *i.e.* the matrix (), is called the
*transposed matrix*
**A**
.

Example. If
,
then
.

(Crystallographers frequently write negative numbers as ,
*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 (..)
.

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**)
, (**b**)
, ..., see also p. ,
List of symbols.

A square matrix **A** is called *symmetric*
if **A**
= **A**,
*i.e.* if holds for any pair .

A symmetric matrix is called a *diagonal matrix*
if for .

A diagonal matrix with all elements is called the
*unit matrix* **I**.

A matrix consisting of
zeroes only, *i.e.* for any pair is called the
**O**-matrix.

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

** Next:** Rules for matrix calculations
**Up:** Matrices and determinants
** Previous:** Motivation
**Copyright © 2002 International Union of
Crystallography**

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