Frobenius matrix

A Frobenius matrix is a special matrix that is used in numerical mathematics . A matrix is ​​a Frobenius matrix if it has the following three properties:

• there are only ones on the main diagonal
• in at most one column there are any entries below the main diagonal
• all other entries are zero

The following matrix is ​​an example:

${\displaystyle A={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&a_{32}&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&a_{n2}&0&\cdots &1\end{pmatrix}}}$

Frobenius matrices always have a determinant of the value 1 and are therefore invertible. Its inverse matrix is formed by changing the sign of all entries outside the main diagonal. So the inverse of the above example is:

${\displaystyle A^{-1}={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&-a_{32}&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&-a_{n2}&0&\cdots &1\end{pmatrix}}}$

This formula can even be generalized to any matrix powers. So applies to everyone : ${\displaystyle k\in \mathbb {Z} }$

${\displaystyle A^{k}={\begin{pmatrix}1&0&0&\cdots &0\\0&1&0&\cdots &0\\0&k\cdot a_{32}&1&\cdots &0\\\vdots &\vdots &\vdots &\ddots &\vdots \\0&k\cdot a_{n2}&0&\cdots &1\end{pmatrix}}=I+k\cdot (A-I)}$

Here stands for the identity matrix . ${\displaystyle I}$

The Frobenius matrices are named after Ferdinand Georg Frobenius . In the description of the Gaussian elimination method, they appear as representation matrices for the Gaussian transformations. If a matrix is ​​multiplied by a Frobenius matrix from the left, then a scalar multiple of a certain line is added to one or more lines below. This corresponds to one of the elementary operations of the Gaussian elimination method (in addition to the operation of interchanging lines and multiplying a line by a scalar multiple).

literature

• Josef Stoer: Numerical Mathematics. An introduction. Taking into account lectures by FL Bauer . Volume 1. 9th edition. Springer, Berlin et al. 2005, ISBN 3-540-21395-3 , p. 201.

Web links

• Frobenius Matrix (Encyclopedia of Mathematics)