Frobenius matrix

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A square matrix is a Frobenius matrix if it has the following three properties:

  • all entries on the main diagonal are one
  • the entries below the main diagonal of one column are arbitrary
  • every other entry is zero

Frobenius matrices are named after Ferdinand Georg Frobenius. An alternative name for this class of matrices is Gauss transformation, after Carl Friedrich Gauss.[1]

The following matrix is an example.

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 are invertible. The inverse of a Frobenius matrix is again a Frobenius matrix. It is equal to the original matrix with changed signs outside the main diagonal. The inverse of the example above is therefore:

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}

[edit] Notes

  1. ^ Golub and Van Loan, p. 95.

[edit] References

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