# Does a matrix with an odd number of eigenvalues with a positive real part have a principal submatrix with 1 or2 eigenvalues with a positive real part?

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.

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.