View source for Matrix Viète theorem

''matrix Vieta theorem''

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201701.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

in terms of the roots: up to sign, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201702.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201703.png" />th [[elementary symmetric function]] of the roots <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201704.png" />. See also [[Viète theorem|Viète theorem]].

Consider now a matrix equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201705.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

where the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201706.png" /> and coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201707.png" /> are square complex matrices. A set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201708.png" /> square matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m1201709.png" /> of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017010.png" /> is called independent if the block [[Vandermonde determinant|Vandermonde determinant]]

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017011.png" /></td> </tr></table>

does not vanish. The matrix Viète theorem gives formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017012.png" /> in terms of quasi-determinants, [[#References|[a3]]], [[#References|[a4]]], involving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017013.png" /> independent solutions of (a2), [[#References|[a1]]], [[#References|[a2]]]. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017014.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017015.png" /> independent solutions of (a2), then

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017016.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017017.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120170/m12017018.png" /></td> </tr></table>

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====
<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>