zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

65F30Other matrix algorithms
65F05Direct methods for linear systems and matrix inversion (numerical linear algebra)
68W30Symbolic computation and algebraic computation
15A21Canonical forms, reductions, classification
13P10Gröbner bases; other bases for ideals and modules
15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
[1] Adams, W. W.; Loustaunau, P.: An introduction to Gröbner bases. (1994) · Zbl 0803.13015