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Difference between revisions of "Matrix Viète theorem"

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m (moved Matrix Viete theorem to Matrix Viète theorem over redirect: accented title)
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<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>
 
<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]].
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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
 
Consider now a matrix equation

Revision as of 18:24, 4 December 2014

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
How to Cite This Entry:
Matrix Viète theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matrix_Vi%C3%A8te_theorem&oldid=35328
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article