Lecture Notes 1: Matrix Algebra
                  Part D: Similar Matrices and Diagonalization
                                                 Peter J. Hammond
                                   minor revision 2020 September 26th
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   1 of 76
   Outline
          Eigenvalues and Eigenvectors
                Real Case
                The Complex Case
                Linear Independence of Eigenvectors
          Diagonalizing a General Matrix
                Similar Matrices
          Properties of Adjoint and Symmetric Matrices
                ASelf-Adjoint Matrix has only Real Eigenvalues
          Diagonalizing a Symmetric Matrix
                Orthogonal Matrices
                Orthogonal Projections
                Rayleigh Quotient
                The Spectral Theorem
          Quadratic Forms and Their Definiteness
                Quadratic Forms
                The Eigenvalue Test of Definiteness
                Sylvester’s Criterion for Definiteness
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   2 of 76
   Definitions in the Real Case
          Definition
          Consider any n ×n matrix A.
          The scalar λ ∈ R is an eigenvalue of A,
          just in case the equation Ax = λx has a non-zero solution.
          In this case the solution x ∈ Rn \ {0} is an eigenvector,
          and the pair (λ,x) is an eigenpair.
          The spectrum of the matrix A is the set SA of its eigenvalues.
          Let SR denote the subset of its real eigenvalues.
                   A
          Let SC denote the subset of its complex eigenvalues,
                   A                 C                 R
          which satisfies S              =SA\S .
                                    A                 A
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   3 of 76
   Summary of Main Properties
          Wewill be demonstrating the following properties:
             1. SR ⊆ S and #S ≤n
                    A         A               A
             2. The number #SC of complex eigenvalues is even,
                                            A         C
                  and the members of SA are complex conjugate pairs λ±µi.
             3. SR = ∅ is possible in case n is even, but not if n is odd.
                    A
             4. In case A is symmetric, one has SC = ∅ and SR = SA.
                                                                        A                   A
          University of Warwick, EC9A0 Maths for Economists                 Peter J. Hammond                   4 of 76