Matrix Viète theorem

matrix Vieta theorem

The standard (scalar) Viète formulas express the coefficients of an equation

(a1)

in terms of the roots: up to sign, is the th elementary symmetric function of the roots . See also Viète theorem.

Consider now a matrix equation

(a2)

where the solutions and coefficients are square complex matrices. A set of square matrices of size is called independent if the block Vandermonde determinant

does not vanish. The matrix Viète theorem gives formulas for in terms of quasi-determinants, [a3], [a4], involving independent solutions of (a2), [a1], [a2]. In particular, if are independent solutions of (a2), then

This theorem generalizes to the case of equations in an arbitrary associative ring (cf. also Associative rings and algebras), with an adequate notion of trace and determinant, see [a1], [a2].

References

[a1]	A. Connes, A. Schwarz, "Matrix Vieta theorem revisited" Lett. Math. Phys. , 39 : 4 (1997) pp. 349–353
[a2]	D. Fuchs, A. Schwarz, "Matrix Vieta theorem" Amer. Math. Soc. Transl. (2) , 169 (1995) pp. 15–22
[a3]	I.M. Gel'fand, D. Krob, A. Lascoux, B. Leclerc, V.S. Redakh, J.Y. Thibon, "Noncomutative symmetric functions" Adv. Math. , 112 (1995) pp. 218–348
[a4]	I.M. Gel'fand, V.S. Redakh, "A theory of noncommutative determinants and characteristic functions of graphs I" Publ. LACIM (Univ. Quebec) , 14 pp. 1–26