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I have a matrix $A$ (not necessarily symmetric) that has a positive even number of eigenvalues with a positive real part (they may be positive or complex). Does that imply that $A$ must have a principal submatrix with an odd number of eigenvalues with a positive real part?

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  • $\begingroup$ Zero is also an even number. So, $-I$ is a counterexample. $\endgroup$
    – user1551
    Commented Aug 2, 2021 at 16:11
  • $\begingroup$ Oh you are right! I forgot to say that with even I mean even and positive. $\endgroup$ Commented Aug 2, 2021 at 17:24

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No. Random counterexample: $$ A=\pmatrix{-1&-2&1\\ 0&-1&-2\\ -2&1&0}. $$ The real parts of its eigenvalues are $-2.2030,0.1015$ and $0.1015$. The real parts of the eigenvalues of the principal $1\times1$ or $2\times2$ submatrices of $A$ are $0,-1$ or $-\frac12$.

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