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Given a $3\times3$ matrix is there a criterion capable of telling whether the matrix has a positive eigenvalue?

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    The trace of a matrix is the sum of the eigenvalues and the determinant is the product. So in the $3\times 3$ case, if either the trace or determinant is positive, there is at least one positive eigenvalue. The converse is not true though. A matrix with -10, 1, 1 on the diagonal, zeros elsewhere, has a negative trace and determinant but two positive eigenvalues.2012-08-09
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    Probably not what you are looking for, but you can actually calculate the eigenvalues using the cubic formula.2012-08-09
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    Thanks, @N.S., I have been working with this approach but thought there may be better ways and, it would seem, there are.2012-08-09
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    As it was pointed, $\det(A)< 0$ always guarantees a positive eigenvalue, anyhow it is not necessarily a necessary condition. And in my opinion, if this doesn't happen, most of the methods posted are not really much easier than simply writing the solutions.2012-08-09

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