4: Eigenvalues, Eigenvectors, Diagonalization

Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1. Theorem 1.2. Let A be an n× n matrix. Then A is invertible if and only if det(A) 6= 0. Exercise 1.3. Let A be an n×n matrix with entries Aij, i, j ∈ {1, . . . , n}, and let Sn denote the set of all permutations on n elements. For σ ∈ Sn, let sign(σ) := (−1) , where σ can be written as a composition of N transpositions. Then