On a family of Laurent polynomials generated by 2x2 matrices

الأصل · EN

To a 2×2 matrix G with complex entries, we relate the sequence of Laurent polynomial Lₙ(z,G)= (G[smallmatrixz&0 0&z⁻¹smallmatrix]G)ⁿ. It turns out that for each n, the family {Lₙ(z,G)}G, where G runs over the set of all 2×2 matrices, is a three-parametric family. A natural parametrization of this family is found. The polynomial Lₙ(z,G) is expressed in terms of these parameters and the Chebyshev polynomial Tₙ. The zero set of the polynomial Lₙ(z,G) is described.