Geodesic
Indefinite numerical range of $3\times 3$ matrices
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 221-239 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $3\times 3$ 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.
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.
Classification : 15A60, 15A63, 46C20
Keywords: indefinite numerical range; indefinite inner product space; plane algebraic curve 
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Bebiano, N.; Providência, J. da; Teixeira, R. Indefinite numerical range of $3\times 3$ matrices. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 1, pp. 221-239. http://geodesic.mathdoc.fr/item/CMJ_2009_59_1_a15/

[1] Ball, W. W. R.: On Newton's classification of cubic curves. Proc. London Math. Soc. 22 (1890), 104-143 \JFM 23.0778.05.

[2] Bebiano, N., Lemos, R., Providência, J. da, Soares, G.: On generalized numerical ranges of operators on an indefinite inner product space. Linear and Multilinear Algebra 52 (2004), 203-233. | DOI | MR

[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. | MR

[4] Brieskorn, E., Knörrer, H.: Plane Algebraic Curves. Birkhäuser Verlag, Basel (1986). | MR

[5] Fiedler, M.: Geometry of the numerical range of matrices. Linear Algebra Appl. 37 (1981), 81-96. | DOI | MR | Zbl

[6] Horn, R. A., Johnson, C. R.: Matrix Analysis. Cambridge University Press, New York (1985). | MR | Zbl

[7] Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis. Cambridge University Press, New York (1991). | MR | Zbl

[8] Kippenhahn, R.: Über den Wertevorrat einer Matrix. Math. Nachr. 6 (1951), 193-228. | DOI | MR | Zbl

[9] Li, C.-K., Rodman, L.: Numerical range of matrix polynomials. SIAM J. Matrix Anal. Appl. 15 (1994), 1256-1265. | DOI | MR | Zbl

[10] Li, C.-K., Rodman, L.: Shapes and computer generation of numerical ranges of Krein space operators. Electr. J. Linear Algebra 3 (1998), 31-47. | MR | Zbl

[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. | MR

[12] Murnaghan, F. D.: On the field of values of a square matrix. Proc. Nat. Acad. Sci. USA 18 (1932), 246-248. | DOI | Zbl

[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. | MR

[14] Nakazato, H., Psarrakos, P.: On the shape of numerical range of matrix polynomials. Linear Algebra Appl. 338 (2001), 105-123. | DOI | MR | Zbl

[15] Psarrakos, P.: Numerical range of linear pencils. Linear Algebra Appl. 317 (2000), 127-141. | DOI | MR | Zbl

[16] Shapiro, H.: A conjecture of Kippenhahn about the characteristic polynomial of a pencil generated by two Hermitian matrices. II. Linear Algebra Appl. 45 (1982), 97-108. | DOI | MR | Zbl