Elementary symmetric of a sum of matrices

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

Given a Matrix $ A $, the $ k $-th 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.

The 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) $.


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