Counter diagonal

Counter diagonals (red) and further counter diagonals (blue) of a (4 × 4) matrix

In mathematics , the counter- diagonal or antidiagonal of a square matrix consists of the matrix elements that lie on an imaginary line running diagonally from top right to bottom left. More generally, the counter- diagonals of a matrix consist of the matrix elements that lie on any diagonal line from top right to bottom left. Occasionally, the opposite diagonals of a matrix are also referred to as "secondary diagonals"; However, the secondary diagonals of a matrix are usually understood to mean the diagonals of the matrix that run parallel to the main diagonal .

The counter diagonals of a matrix are, for example, usually used by Sarrus .

definition

The opposite diagonal of a square matrix

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

consists of those entries in the matrix that lie on the diagonal from top right to bottom left, i.e. the elements

${\displaystyle a_{1,n},a_{2,n-1},\ldots ,a_{n-1,2},a_{n,1}}$.

The opposite diagonal thus consists of those matrix entries whose sum of the row and column index results in the value . In general, the counter-diagonals of a matrix of any size consist of those entries whose sum of row and column index is constant, i.e. for which ${\displaystyle a_{i,j}}$ ${\displaystyle n+1}$

${\displaystyle i+j=k}$

with a constant value applies. ${\displaystyle k}$

example

The opposite diagonal of the real matrix

${\displaystyle A={\begin{pmatrix}0&1&2&3\\1&2&3&4\\2&3&4&5\\3&4&5&6\end{pmatrix}}}$

consists of the elements . The other opposite diagonals of the matrix also have constant entries. Matrices with this property are called Hankel matrices . ${\displaystyle 3,3,3,3}$

use

Diagonals and counter-diagonals in the rule of Sarrus

The opposite diagonal of a matrix is ​​the counterpart to the diagonal of the matrix, which is also referred to as the main diagonal . The opposite diagonals of a matrix are then the counterparts to the diagonals of the matrix, which outside the main diagonals are also referred to as secondary diagonals of the matrix.

In Sarrus' rule , the determinant of a matrix is ​​calculated with the help of the diagonals and counter-diagonals of the matrix extended by the first two columns. ${\displaystyle 3\times 3}$

A matrix that is symmetric with respect to its counter-diagonal is called a persymmetric matrix . A Toeplitz matrix is a persymmetrical matrix in which the entries on the main diagonal and the secondary diagonals are constant. A matrix that is symmetrical both with respect to its diagonal and with respect to its counter-diagonal is called a bisymmetrical matrix .

literature

• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 978-0-521-83940-2 .

Individual evidence

1. ^ Roger A. Horn, Charles Johnson: Matrix analysis . Cambridge University Press, 2013, pp. 33 .

Web links

• Eric W. Weisstein : Skew diagonal . In: MathWorld (English).