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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

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  • $\begingroup$ Sometimes one arrives at this situation from constructing an isomorphism from a high-dimensional matrix algebra with symmetric generators to a low-dimensional one. Is it your setting by any chance? $\endgroup$
    – Dima Pasechnik
    Commented Dec 14, 2018 at 15:34

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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.

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