Main diagonal

Main diagonal (red) and secondary diagonals (blue) of a (4 × 4) matrix

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

${\displaystyle A={\begin{pmatrix}a_{1,1}&a_{1,2}&\ldots &a_{1,n}\\a_{2,1}&a_{2,2}&\ldots &a_{2,n}\\\vdots &\vdots &&\vdots \\a_{m,1}&a_{m,2}&\ldots &a_{m,n}\end{pmatrix}}}$

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. ${\displaystyle a_{k,k}}$ ${\displaystyle 1\leq k\leq \min\{m,n\}}$ ${\displaystyle a_{i,j}}$

Examples

The main diagonal of the matrix

${\displaystyle A={\begin{pmatrix}1&4&2&1\\3&0&3&1\\5&1&2&0\\4&1&2&3\end{pmatrix}}}$

consists of the four entries and . ${\displaystyle 1,0,2}$ ${\displaystyle 3}$

The following 3 matrices have their main diagonal marked with red ones.

${\displaystyle {\begin{pmatrix}\color {red}{1}&0&0\\0&\color {red}{1}&0\\0&0&\color {red}{1}\end{pmatrix}}\qquad {\begin{pmatrix}\color {red}{1}&0&0&0\\0&\color {red}{1}&0&0\\0&0&\color {red}{1}&0\end{pmatrix}}\qquad {\begin{pmatrix}\color {red}{1}&0&0\\0&\color {red}{1}&0\\0&0&\color {red}{1}\\0&0&0\end{pmatrix}}}$

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. ${\displaystyle (2\times 2)}$ ${\displaystyle (3\times 3)}$

See also

• Diagonal dominant matrix

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).