Linear matrix representations of ternary forms

Mao-Ting Chien

Department of Mathematics

Soochow University

mtchien@scu.edu.tw

    Peter Lax (1958) conjectured that every hyperbolic ternary form `F(t, x, y)` of degree `n` admits a determinantal linear matrix representation, i.e., there exist `n \times n` real symmetric matrices `H` and `K` satisfying `F(t,x,y)=\det(tI_n+xH+yK)`. Helton and Vinnikov confirmed in 2007 the Lax conjecture is true. In this talk, we study the linear matrix representations of the hyperbolic ternary forms associated to some matrices.

Keyword: Hyperbolic ternary form, determinantal representation, Lax conjecture

References

[1] Mao-Ting Chien, Hiroshi Nakazato, Unitary similarity of the determinantal representation of unitary bordering matrices, Linear Algebra Appl., 541(2018), 13-35.
[2] Mao-Ting Chien, Hiroshi Nakazato, Symmetric representation of ternary forms associated to some Toeplitz matrices, Symmetry, 10(2018), 55; doi:10.3390/sym10030055