Indefinite numerical range of 3 × 3 matrices

N. Bebiano; J. da Providência; R. Teixeira

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 1, page 221-239
  • ISSN: 0011-4642

Abstract

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The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the 3 × 3 indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range.

How to cite

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Bebiano, N., Providência, J. da, and Teixeira, R.. "Indefinite numerical range of $3\times 3$ matrices." Czechoslovak Mathematical Journal 59.1 (2009): 221-239. <http://eudml.org/doc/37919>.

@article{Bebiano2009,
abstract = {The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the $3\times 3$ indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range.},
author = {Bebiano, N., Providência, J. da, Teixeira, R.},
journal = {Czechoslovak Mathematical Journal},
keywords = {indefinite numerical range; indefinite inner product space; plane algebraic curve; indefinite numerical range; indefinite inner product space; plane algebraic curve; krein spaces},
language = {eng},
number = {1},
pages = {221-239},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Indefinite numerical range of $3\times 3$ matrices},
url = {http://eudml.org/doc/37919},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Bebiano, N.
AU - Providência, J. da
AU - Teixeira, R.
TI - Indefinite numerical range of $3\times 3$ matrices
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 221
EP - 239
AB - The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler’s approach for definite inner product spaces. The classification of the associated curve is presented in the $3\times 3$ indefinite case, using Newton’s classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range.
LA - eng
KW - indefinite numerical range; indefinite inner product space; plane algebraic curve; indefinite numerical range; indefinite inner product space; plane algebraic curve; krein spaces
UR - http://eudml.org/doc/37919
ER -

References

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  3. Bebiano, N., Lemos, R., Providência, J. da, Soares, G., On the geometry of numerical ranges in spaces with an indefinite inner product, Linear Algebra Appl. 399 (2005), 17-34. (2005) MR2152407
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  11. Li, C.-K., Tsing, N. K., Uhlig, F., Numerical ranges of an operator in an indefinite inner product space, Electr. J. Linear Algebra 1 (1996), 1-17. (1996) MR1401906
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  13. Nakazato, H., Bebiano, N., Providência, J. da, The J -Numerical Range of a J -Hermitian Matrix and Related Inequalities, Linear Algebra Appl. 428 (2008), 2995-3014. (2008) MR2416604
  14. Nakazato, H., Psarrakos, P., 10.1016/S0024-3795(01)00374-3, Linear Algebra Appl. 338 (2001), 105-123. (2001) Zbl0996.15018MR1861116DOI10.1016/S0024-3795(01)00374-3
  15. Psarrakos, P., 10.1016/S0024-3795(00)00145-2, Linear Algebra Appl. 317 (2000), 127-141. (2000) Zbl0966.15014MR1782206DOI10.1016/S0024-3795(00)00145-2
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