In matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A. If A is a differentiable map from the real numbers to n × n matrices,
Equivalently, if dA stands for the differential of A, the formula is
It is named after the mathematician C.G.J. Jacobi.
We first prove a preliminary lemma:
Lemma. Let A and B be a pair of square matrices of the same dimension n. Then
Proof. The product AB of the pair of matrices has components
Replacing the matrix A by its transpose AT is equivalent to permuting the indices of its components:
The result follows by taking the trace of both sides:
Theorem. (Jacobi's formula) For any differentiable map A from the real numbers to n × n matrices,
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:
so that, by the chain rule, its differential is
This summation is performed over all n×n elements of the matrix.
To find ∂F/∂Aij consider that on the right hand side of Laplace's formula, the 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 ∂ / ∂Aij:
Thus, by the product rule,
Now, if an element of a matrix Aij and a cofactor adjT(A)ik of element Aik lie on the same row (or column), then the cofactor will not be a function of Aij, because the cofactor of Aik is expressed in terms of elements not in its own row (nor column). Thus,
All the elements of A are independent of each other, i.e.
where δ is the Kronecker delta, so
and applying the Lemma yields
For any invertible matrix A, the inverse A−1 is related to the adjugate by A−1 = (det A)−1 adj A. It follows that if A(t) is invertible for all t, then
which can be alternatively written as
Furthermore, taking A(t ) = exp(tB ) in the first equation, we obtain
A quicker proof of Jacobi's formula is as follows. By the chain rule, we have
where (:) denotes tensor double-contraction.
- Magnus & Neudecker (1999), Part Three, Section 8.3