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An application of the Gröbner basis in computation for the minimal polynomials and inverses of block circulant matrices. (English) Zbl 1005.65040
The authors show that the ring $F[\tau_1,\ldots,\tau_n]$ of all level-$n(r_1,\ldots,r_n)$-block circulant matrices over the field $F$ is isomorphic to $F[x_1,\ldots,x_n]/<x_1^{k_1}-r_1,\ldots, x_n^{k_n}-r_n>$, 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 $f(\tau_1,\ldots,\tau_n)$ is the monic polynomial that generates the ideal $<x_1^{k_1}-r_1,\ldots,x_n^{k_n}-r_n,y-f(x_1,\ldots,x_n)> \cap F[y]$, which can be calculated readily using Gröbner basis techniques. A similar formula is given for the annihilation ideal of a set of level-$n(r_1,\ldots,r_n)$-block circulant matrices. The authors show that the matrix $f(\tau_1,\ldots\tau_n)$ is non-singular if and only if $$1 \in \langle f(x_1,\dots,x_n),x_1^{k_1}-r_1,\dots,x_n^{k_n}-r_n\rangle,\text{ i.e. }1 = fg + \sum w_i(x_i^{k_i}-r_i).$$ 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 $n(r_1,\ldots,r_n)$-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 15B33 Matrices over special rings (quaternions, finite fields, etc.)
CoCoA
Full Text:
##### References:
 [1] Adams, W. W.; Loustaunau, P.: An introduction to Gröbner bases. (1994) · Zbl 0803.13015