Determinant derivatives

(D.1) |

where is the minor of the matrix respect to the element and therefore does not depend explicitly by the elements of the raw . So the derivative with respect to will be:

(D.2) |

and for the logarithmic derivative we have:

We want to find a simple relation to evaluate second derivatives of the determinant. We write the relation

if we derive this equation for we obtain:

where we substitute the with the eq. D.4. Because of this equation is zero for all this means that the expression in parentheses is zero, and this yields to: