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For an experiment I'm running I want to be able to generate random real $n\times{}n$ matrices that have complex eigenvalues with only negative real part.

Is there any way of constructing these?

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Actually this turns out to be very simple.

  1. Generate a random $n\times{}n$ matrix $\bf A$.
  2. Construct the eigendecomposition $\bf A = \bf Q \bf \Lambda \bf{Q^{-1}}$.
  3. Set $\bf \Lambda^-$ identical to $\bf \Lambda$, but with the real part of each element set to have negative sign.
  4. Reconstruct $\bf A^- = \bf Q \bf \Lambda^- \bf {Q^{-1}}$.
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