Lecture 3 : Proof of Burton , Pemantle Theorem

It is easy to see that for any pair of matrices A ∈ Rn×k and B ∈ Rk×n, Tr(AB) = Tr(BA). The matrix dot product is defined analogous to the vector dot product. For matrices A,B ∈ Rk×n, A •B := k ∑ i=1 n ∑ j=1 A(i, j)B(i, j). It is easy to see that A •B = Tr(AB). Lemma 3.1. For any symmetric matrix A with eigenvalues λ1, . . . , λn, Tr(A) = n ∑ i=1 λi. Proof. Let x1, . . . , xn be orthonormal eigenvectors corresponding to λ1, . . . , λn. Also, let 1i be the indicator vector of the i-th coordinate. Then, by definition of the trace, Tr(A) = n ∑ i=1 1iA1i