Jacobi's formula

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In matrix calculus, Jacobi's formula expresses the differential of the determinant of a matrix A in terms of the adjugate of A and the differential of A. The formula is

It is named after the mathematician C.G.J. Jacobi.

[edit] Derivation

We first prove a preliminary lemma:

Lemma. Given a pair of square matrices A and B of the same dimension n, then

Proof. The product AB of the pair of matrices has components

Replacing the matrix A by its transpose A^T is equivalent to permuting the indices of its components:

The result follows by taking the trace of both sides:

Theorem.

Proof. Laplace's formula for the determinant of a matrix A can be stated as

Notice that the summation is performed over some arbitrary row i of the matrix.

The determinant of A can be considered to be a function of the elements of A:

det(A) = F(A₁₁,A₁₂,...,A₂₁,A₂₂,...,A_nn)

so that its differential is

This summation is performed over all n×n elements of the matrix.

To find ∂F / ∂A_ij consider that in the right side of Laplace's formula, index i can be chosen at will (in order to optimize calculations: any other choice would eventually yield the same result, but it could be much harder). In particular, it can be chosen to match the first index of ∂ / ∂A_ij:

Now, if an element of a matrix A_ij and a cofactor adj^T(A)_ik of element A_ik lie on the same row (or column), then the cofactor will not be a function of A_ij, because the cofactor of A_ik is expressed in terms of elements not in its own row (nor column). Thus,

so

All the elements of A are independent of each other, i.e.

where δ is the Kronecker delta, so

Therefore,

and applying the Lemma yields