Is there an explicit formula for the derivative of the matrix minor?

Question

Let $M_n$ be the vector space of real $n \times n$ real matrices. Fixe $1 < k < n$, and let $I=(i_1,i_2, \dots, i_{k}),J=(j_1, \dots, j_{k})$ be multi-indices, i.e. $1 \le i_1 < i_2 < \dots < i_{k} \le n $ and $1 \le j_1 < i_2 < \dots < j_{k} \le n $.

Let $M^I_J:M_n \to \mathbb{R}$ be the function which assigns every matrix its $(I,J)$ minor.

Question: Is there an explicit formula for the gradient of $M^I_J$?

That is, given $A \in M_n$, what is $(\nabla M^I_J)(A)$?

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For $k=n$, $M^I_J$ is the determinant, and $(\nabla \det)(A)=\Cof(A)$ is the cofactor matrix of $A$.

Is there an explicit generalization of Jacobi formula for minors of other degrees?

Answer 1

Denote by $\mathbf{A}_{ij}$ the submatrix obtained by removing the ith row and the jth column of $\mathbf{A}$.

You are interested in differentiating the scalar-valued function $\phi(\mathbf{A}) = \det(\mathbf{A}_{ij})$. It holds $$ \frac{\partial \phi}{\partial \mathbf{A}_{ij}} = \phi \cdot \mathbf{A}_{ij}^{-T} $$ the other terms are null, for instance $$ \frac{\partial \phi}{\partial A_{ij}} = 0 $$ It remains to reallocate these derivative terms (right side) at the right place...