Given a symmetric matrix $A$, which has complex values in the diagonal, but whose all other entries are real, I am interested in finding an orthonormal transformation $Q$ such that $Q^tAQ$ is a pentadiagonal matrix whose entries outside of the main diagonal are real. So far, I have not even been able to prove if it's possible to always find such a transformation. Is there maybe any constructive numerical algortihm to attain this?

  • $\begingroup$ what happens with 6 by 6 matrix? $\endgroup$
    – Will Jagy
    Mar 25, 2021 at 18:18
  • $\begingroup$ I really don't know. What happens? Anyway, in the problem I am concerned with the matrices are much larger. $\endgroup$
    – Qwertuy
    Mar 25, 2021 at 19:17
  • $\begingroup$ I imagine matrix $Q$ needs to have real coefficients ? $\endgroup$
    – Pohoua
    Mar 26, 2021 at 0:29


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