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Hurwitz Matrices require that for a polynomial to have all negative real part roots then the determinant of the principle minors must be positive.

I know that if we have a positive definite matrix then all the real parts of the eigenvalues are positive, and also that the determinant of the principle minors are positive.

My question is: How are the two related?

As the characteristic polynomial to the Hurwitz matrix will have positive real parts to the solutions. So what is the relation (or formation of Hurwitz matrices) between the polynomial of negative real part roots and the Hurwitz matrix, which will have a characteristic polynomial of positive real part roots.

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