Functions of Matrices and
                   ResearchMatters
                Nearest Correlation Matrices
                     February 25, 2009
                            Nick Higham
                        Nick Higham
                      School of Mathematics
                    Director of Research
                   TheUniversity of Manchester
                   School of Mathematics
                      higham@ma.man.ac.uk
             http://www.ma.man.ac.uk/~higham/
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                                                                                                                                                                                                                                                                                                                                                                                                                                   It is
                                                                                                                                                                                                                                                                                                                                                                                                                                         −1
                                                                                                                                                                                                                                                                                                                                                                                                                                       A ,
                                                                                                                                                                                                                                                                                                                                                                                                                                       eA,
                                                                                                                                                                                                                                                                                                                                                                                                                                       √
                                                                                                                                                                                                                                                                                                                                                                                                                                         A,
                                                                                                                                                                                                                                                                                                                                                                                                                                       . . .
   Whatis a Matrix Function?
     It’s not
          det(A) or trace(A),
          elementwise evaluation: f(aij),
            T
          A ,
          matrix factor (e.g., A = LU).
  MIMS              Nick Higham                 Matrix Functions      2/ 33
   Whatis a Matrix Function?
     It’s not
          det(A) or trace(A),
          elementwise evaluation: f(aij),
            T
          A ,
          matrix factor (e.g., A = LU).
     It is
            −1
          A ,
          eA,
          √
            A,
          . . .
  MIMS              Nick Higham                 Matrix Functions      2/ 33
   Cayley and Sylvester
          Term “matrix” coined in 1850
          by James Joseph Sylvester,
          FRS(1814–1897).
          Matrix algebra developed by
          Arthur  Cayley,   FRS (1821–
          1895).
          Memoir on the Theory of Ma-
          trices (1858).
  MIMS            Nick Higham                Matrix Functions     3/ 33