Difference between revisions of "Matrix Viète theorem"
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''matrix Vieta theorem'' | ''matrix Vieta theorem'' | ||
The standard (scalar) Viète formulas express the coefficients of an equation | 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, | + | 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|Viète theorem]]. |
Consider now a matrix equation | 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 | + | 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|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 | + | does not vanish. The matrix Viète theorem gives formulas for $A_i$ in terms of quasi-determinants, [[#References|[a3]]], [[#References|[a4]]], involving $n$ independent solutions of (a2), [[#References|[a1]]], [[#References|[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|Associative rings and algebras]]), with an adequate notion of trace and determinant, see [[#References|[a1]]], [[#References|[a2]]]. | This theorem generalizes to the case of equations in an arbitrary associative ring (cf. also [[Associative rings and algebras|Associative rings and algebras]]), with an adequate notion of trace and determinant, see [[#References|[a1]]], [[#References|[a2]]]. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> A. Connes, A. Schwarz, "Matrix Vieta theorem revisited" ''Lett. Math. Phys.'' , '''39''' : 4 (1997) pp. 349–353</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> D. Fuchs, A. Schwarz, "Matrix Vieta theorem" ''Amer. Math. Soc. Transl. (2)'' , '''169''' (1995) pp. 15–22</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> 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</td></tr></table> |
Revision as of 16:56, 1 July 2020
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 |
| [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 |
How to Cite This Entry:
Matrix Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_Vi%C3%A8te_theorem&oldid=50161
Matrix Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_Vi%C3%A8te_theorem&oldid=50161
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article