Matrix Viète theorem

matrix Vieta theorem

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

\begin{equation} \tag{a1} x ^ { n } + a _ { 1 } x ^ { n - 1 } + \ldots + a _ { n - 1 } x + a _ { n } = 0 \end{equation}

in terms of the roots: up to sign, $a_i$ is the $i$th elementary symmetric function of the roots $\alpha_{1} , \ldots , \alpha _ { n }$. See also Viète theorem.

Consider now a matrix equation

\begin{equation} \tag{a2} X ^ { n } + A _ { 1 } X ^ { n - 1 } + \ldots + A _ { n - 1 } X + A _ { n } = 0, \end{equation}

where the solutions $X$ and coefficients $A_i$ are square complex matrices. A set of $n$ square matrices $X _ { 1 } , \ldots , X _ { n }$ of size $m \times m$ is called independent if the block Vandermonde determinant

\begin{equation*} \operatorname { det } \left( \begin{array} { c c c } { I } & { \ldots } & { I } \\ { X _ { 1 } } & { \ldots } & { X _ { n } } \\ { \vdots } & { \ldots } & { \vdots } \\ { X _ { 1 } ^ { n - 1 } } & { \ldots } & { X _ { n } ^ { n - 1 } } \end{array} \right) \end{equation*}

does not vanish. The matrix Viète theorem gives formulas for $A_i$ in terms of quasi-determinants, [a3], [a4], involving $n$ independent solutions of (a2), [a1], [a2]. In particular, if $X _ { 1 } , \ldots , X _ { n }$ are $n$ independent solutions of (a2), then

\begin{equation*} \operatorname { Tr } ( X _ { 1 } ) + \ldots + \operatorname { Tr } ( X _ { n } ) = - \operatorname { Tr } ( A _ { 1 } ), \end{equation*}

\begin{equation*} \operatorname { det } ( X _ { 1 } ) \ldots \operatorname { det } ( X _ { n } ) = ( - 1 ) ^ { n } \operatorname { det } ( A _ { n } ) , \operatorname { det } ( I - \lambda X _ { 1 } ) \ldots \operatorname { det } ( I - \lambda X _ { n } )= \end{equation*}

\begin{equation*} = \operatorname { det } ( 1 + A _ { 1 } \lambda + \ldots + A _ { n } \lambda ^ { n } ). \end{equation*}

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 Zbl 0874.15010
[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