A note on companion matrices

@article{Fiedler2003ANO,
  title={A note on companion matrices},
  author={Miroslav Fiedler},
  journal={Linear Algebra and its Applications},
  year={2003},
  volume={372},
  pages={325-331}
}
  • M. Fiedler
  • Published 1 October 2003
  • Mathematics
  • Linear Algebra and its Applications
A new family of companion forms of polynomial matrices
In this paper a new family of companion forms associated to a regular polynomial matrix is presented. Similar results have been presented in a recent paper by M. Fiedler, where the scalar case is
A new kind of companion matrix
A new kind of companion matrix is introduced, for polynomials of the form c(I») = I»a(I»)b(I»)+c_0, where upper Hessenberg companions are known for the polynomials a(I») and b(I»). This construction
A FACTORIZATION OF THE INVERSE OF THE SHIFTED COMPANION MATRIX
TLDR
A method for factoring the shifted companion matrix CI is presented and the cost of multiplying a vector by (CI) 1 is shown to be O(n), and therefore, any shift and invert eigenvalue method can be implemented efficiently.
2 Characterization of Matrices in H n with a Pentadiagonal Form
TLDR
This work determines which of the Fiedler companion matrices are permutationally similar to a pentadiagonal matrix and describes how to find the permutation involved.
A note on companion pencils
Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding
...
...

References

SHOWING 1-5 OF 5 REFERENCES
Structure ranks of matrices
Matrix analysis
TLDR
This new edition of the acclaimed text presents results of both classic and recent matrix analyses using canonical forms as a unifying theme, and demonstrates their importance in a variety of applications.
Applications of the Companion Matrix
Congenial matrices
  • Linear Algebra Appl
  • 1981