Main diagonal
In mathematics, the main diagonal of a matrix consists of those elements of the matrix that lie on an imaginary line running diagonally from the top left at 45 ° to the bottom right. The diagonals of the matrix that run parallel to the main diagonal are referred to as the secondary diagonals of the matrix. The diagonal of a matrix, which instead runs from top right to bottom left, is called the opposite diagonal of the matrix.
definition
The main diagonal of a matrix
consists of those entries in the matrix that are on the diagonal from top left to bottom right. These are the entries with , i.e. precisely those matrix entries in which the row and column index match.
Examples
The main diagonal of the matrix
consists of the four entries and .
The following 3 matrices have their main diagonal marked with red ones.
use
A square matrix in which only the elements on the main diagonal are different from zero is called a diagonal matrix . If all of these elements have the value one, the so-called identity matrix results . The sum of the main diagonal elements is called the trace of the matrix. A symmetrical matrix is a matrix that is symmetrical about its main diagonal.
The term "secondary diagonal" has no uniform definition and can refer to a diagonal that runs parallel to the main diagonal above or below it, or to one of the counter- diagonals of the matrix that run from top right to bottom left.
The main diagonal plays a special role in calculating the determinant of a matrix. With a matrix, the determinant is the product of the main diagonal elements minus the product of the counter diagonal elements. In the case of a matrix, the determinant can be calculated using Sarrus' rule , in which the main, secondary and counter diagonals are considered. With a triangular matrix of any size, the determinant results directly as the product of the main diagonal elements.
See also
literature
- Christian Voigt, Jürgen Adamy: Collection of formulas for matrix calculation . Oldenbourg Wissenschaftsverlag, Munich 2007, ISBN 978-3-486-58350-2 , p. 19 .
- Peter Gabriel: Matrices, Geometry, Linear Algebra . Birkhäuser Verlag, Berlin / Basel / Boston 1996, ISBN 978-3-7643-5376-6 , pp. 475 .
Web links
- Eric W. Weisstein : Main Diagonal . In: MathWorld (English).