Frobeniusmatrix

A Frobeniusmatrix is a special matrix of the mathematical subfield of numerical analysis. A matrix is ​​a Frobeniusmatrix if it has the following three characteristics:

• On the main diagonal are all ones
• In more than one column under the main diagonal any entries
• All other entries are zero

An example represents the following matrix

Frobeniusmatrizen have a determinant of the magnitude 1 and are thus invertible. Its inverse matrix is ​​formed by the sign of all entries is changed outside the main diagonal. The inverse of the above example is calculated thus:

The Frobeniusmatrizen are named after Ferdinand Georg Frobenius. They occur in the description of the Gaussian elimination method on the representation matrices of the Gaussian transformations.

Is a matrix of left multiplied by a Frobeniusmatrix then a scalar multiple of a given row is added to one or more rows below. The multiplication by the inverse of a Frobeniusmatrix yields the corresponding subtraction of scalar multiples of a line. This corresponds to one of the elementary operations of Gaussian elimination ( in addition to the operation of the permutation of rows and multiplication of a row with a scalar multiple ).