Q: Efficient eigen-decomposition of a real matrix with all real eigenvalues I'm optimising a radar algorithm that results in real matrices which are not symmetric but which are guaranteed to have real eigenvalues. Each matrix is therefore similar to a symmetric matrix. I am solving for the eigenvalues using the LAPACK general real eigensolver SGEEV which produces the correct eigenvalues with zero imaginary component within numerical accuracy. This is obviously inefficient. What I want to do is apply a similarity transformation to these special real matrices to reduce them to symmetric form (or tri-diagonal symmetric) and then call a symmetric eigensolver. Any ideas how to do this? Thanks

A: On computation of real eigenvalues of matrices via the Adomian decomposition

A new approach based on the Adomian decomposition method and the
  Faddeev-Leverrier's algorithm is presented for finding real
  eigenvalues of any desired real matrices. The method features accuracy
  and simplicity. In contrast to many previous techniques which merely
  afford one specific eigenvalue of a matrix, the method has the
  potential to provide all real eigenvalues. Also, the method does not
  require any initial guesses in its starting point unlike most of
  iterative techniques.

