Elementary symmetric of a sum of matrices

Importance: High ✭✭✭
Subject: Algebra
Recomm. for undergrads: no
Posted by: rscosa
on: December 8th, 2008

\begin{problem} % Enter your conjecture in LaTeX Given a Matrix $A$, the $k$-th \Def{elementary symmetric function} of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by-$k$ principal minors.

Find a closed expression for the $k$-th elementary symmetric function of a sum of N $n$-by-$n$ matrices, with $0\le N\le k\le n$ by using partitions.


% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}

The \Def{Newton-Girard formulas} imply particular expressions for small values of $k$ and $N$, for example, $S_2(A+B)=S_2(A)+S_2(B)+S_1(A)S_1(B)-S_1(AB)$.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.

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