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,...

We present a method for calculating some select eigenvalues and corresponding eigenvectors of a given Hamiltonian. We show that it is possible to target the eigenvalues and eigenvectors of interest without diagonalizing the full Hamiltonian, by using any arbitrary physical property of the eigenvectors. This allows us to target, for example, the eigenvectors based on their localization propertie...