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I have a real matrix $A$ (not necessarily symmetric) with an odd number (greater than 1) of eigenvalues with a positive real part. Does this imply that $A$ has a principal submatrix with only one eigenvalue with a positive real part, or an even number of eigenvalues with a positive real part?

This is clearly true for a $3\times 3$ matrix (In fact, both things happen in this case), but I don't know if this can be generalized even further.

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This is obviously true, because the number of eigenvalues with positive real parts in every principal $2\times2$ submatrix is either $0,1$ or $2$.

If the word "even" in you question refers to a positive even number, then the statement is false. An easy counterexample is given by the $5\times5$ permutation matrix for a $5$-cycle. The matrix has three eigenvalues with positive real parts and each of its smaller principal submatrices is nilpotent.

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