Linear Algebra Theorem Reference Highlights from Matrix Analysis by Horn & Johnson

1.1.6 Let p(•) be any polynomial. If λ ∈ σ(A) with associated eigenvector y, then p(λ) ∈ σ(p(A)) also with associated eigenvector y. 1.1.p1 λ ∈ σ(A)⇔ λ−1 ∈ σ(A−1). 1.2.4-e If T ∈Mn is triangular, then σ(A) = {t11, ..., tnn}. 1.2.6-e If A ∈Mn(R) and n is odd, A has at least one real eigenvalue. 1.2.5 Ek(A) is the sum of the principle minors (the square k × k submatrices lying along the diagonal). 1.2.9 Definition Sk(λ1, λ2, ..., λn) ≡ ∑ 1≤i1<...<ik≤n k ∏