Hadamard
                                                                               Matrices:
                                                                               Truth &
                         Hadamard Matrices:                                   Consequences
                                                                             Raymond Nguyen
                        Truth & Consequences                                 Advisor: Peter
                                                                               Casazza
                                                                             The University of
                                                                               Missouri
                                                                              Math 8190
                                                                            (Master’s Project)
                              Raymond Nguyen                                Basic Theory of
                                                                            Hadamard
                           Advisor: Peter Casazza                           Matrices
                          The University of Missouri                        Hadamard Matrix
                                                                            Constructions
                        Math 8190 (Master’s Project)                        Applications of
                                                                            Hadamard
                                                                            Matrices
                                 Spring 2020
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                                                                     Hadamard
  Outline                                                             Matrices:
                                                                      Truth &
                                                                     Consequences
                                                                    Raymond Nguyen
                                                                     Advisor: Peter
                                                                      Casazza
                                                                    The University of
                                                                      Missouri
   1 Basic Theory of Hadamard Matrices                               Math 8190
                                                                    (Master’s Project)
                                                                   Basic Theory of
                                                                   Hadamard
   2 Hadamard Matrix Constructions                                 Matrices
                                                                   Hadamard Matrix
                                                                   Constructions
                                                                   Applications of
   3 Applications of Hadamard Matrices                             Hadamard
                                                                   Matrices
                                                                         2/55
                                                                                                                                         Hadamard
   Table of Contents                                                                                                                      Matrices:
                                                                                                                                          Truth &
                                                                                                                                        Consequences
                                                                                                                                      Raymond Nguyen
                                                                                                                                        Advisor: Peter
     1    Basic Theory of Hadamard Matrices                                                                                               Casazza
                                                                                                                                      The University of
                                                                                                                                          Missouri
              Definition & Examples                                                                                                       Math 8190
                                                                                                                                      (Master’s Project)
              Properties                                                                                                             Basic Theory of
                                                                                                                                     Hadamard
                                                                                                                                     Matrices
                                                                                                                                      Definition & Examples
              The Hadamard Conjecture                                                                                                 Properties
                                                                                                                                      The Hadamard Conjecture
                                                                                                                                     Hadamard Matrix
                                                                                                                                     Constructions
     2    Hadamard Matrix Constructions                                                                                              Applications of
                                                                                                                                     Hadamard
                                                                                                                                     Matrices
     3    Applications of Hadamard Matrices
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                                                                                                                                         Hadamard
   What is a Hadamard Matrix?                                                                                                             Matrices:
                                                                                                                                          Truth &
                                                                                                                                        Consequences
                                                                                                                                      Raymond Nguyen
                                                                                                                                        Advisor: Peter
     Definition (Hadamard Matrix)                                                                                                          Casazza
                                                                                                                                      The University of
     A square matrix H of order n whose entries are +1 or −1 is called a                                                                  Missouri
                                                                                                                                         Math 8190
     Hadamard matrix of order n provided its rows are pairwise orthogonal –                                                           (Master’s Project)
     i.e.,                                                                                                                           Basic Theory of
                                                             T                                                                       Hadamard
                                                       HH =n·In:                                                           (1)       Matrices
                                                                                                                                      Definition & Examples
                                                                                                                                      Properties
                                                                                                                                      The Hadamard Conjecture
     Note that (1) implies that H has an inverse 1HT. Consequently, its                                                              Hadamard Matrix
     columns are also pairwise orthogonal – i.e., n                                                                                  Constructions
                                                                                                                                     Applications of
                                                                                                                                     Hadamard
                                                       HTH =n·I :                                                          (2)       Matrices
                                                                           n
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