*(English)*Zbl 1005.65040

The authors show that the ring $F[{\tau}_{1},...,{\tau}_{n}]$ of all level-$n({r}_{1},...,{r}_{n})$-block circulant matrices over the field $F$ is isomorphic to $F[{x}_{1},...,{x}_{n}]/<{x}_{1}^{{k}_{1}}-{r}_{1},...,{x}_{n}^{{k}_{n}}-{r}_{n}>$, where the dimension of the ${i}^{th}$ block is ${k}_{i}$ and ${r}_{i}$ is the field element in the lower left corner of the ${i}^{th}$ block. They further show that the minimal polynomial of the matrix $f({\tau}_{1},...,{\tau}_{n})$ is the monic polynomial that generates the ideal $<{x}_{1}^{{k}_{1}}-{r}_{1},...,{x}_{n}^{{k}_{n}}-{r}_{n},y-f({x}_{1},...,{x}_{n})>\cap F\left[y\right]$, which can be calculated readily using Gröbner basis techniques. A similar formula is given for the annihilation ideal of a set of level-$n({r}_{1},...,{r}_{n})$-block circulant matrices.

The authors show that the matrix $f({\tau}_{1},...{\tau}_{n})$ is non-singular if and only if

They also show how Gröbner basis techniques can be used to calculate $g$, the inverse of $f$, explicitly.

Finally, the authors give two algorithms, both using Gröbner bases, for the inverse of a level $n({r}_{1},...,{r}_{n})$-block circulant matrix over a quaternion division algebra. All algorithms are explained in detail and have been implemented by the authors in CoCoa 4.0, a computer algebra system.

##### MSC:

65F30 | Other matrix algorithms |

65F05 | Direct methods for linear systems and matrix inversion (numerical linear algebra) |

68W30 | Symbolic computation and algebraic computation |

15A21 | Canonical forms, reductions, classification |

13P10 | Gröbner bases; other bases for ideals and modules |

15A33 | Matrices over special rings |