= Jacobi's formula
{c}
{title2=$D(\det)_A(H)=\operatorname{tr}(\operatorname{adj}(A)H)$}
{wiki}
= Jacobi formula
{c}
{synonym}
For an arbitrary square <matrix> $A$, the <derivative of the determinant> in direction $H$ is the displayed expression. Differentiating the <determinant>'s <multilinear> expansion gives the sum of entries of $H$ weighted by their cofactors, equal to the displayed <trace>. If $A$ is invertible, $\operatorname{adj}(A)=(\det A)A^{-1}$ gives $(\det A)\operatorname{tr}(A^{-1}H)$, and division by $\det A$ gives the <Jacobi determinant derivative formula> for the logarithmic <derivative>. The adjugate expression remains valid at singular <matrices>.
Back to article page
