Jacobi's formula

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, then

Equivalently, if dA stands for the differential of A, the general formula is

Lemma. Let A and B be a pair of square matrices of the same dimension n. Then

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

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:

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:

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,

Proof. Using the definition of a directional derivative together with one of its basic properties for differentiable functions, we have

The following is a useful relation connecting the trace to the determinant of the associated matrix exponential:

This statement is clear for diagonal matrices, and a proof of the general claim follows.

The desired result follows as the solution to this ordinary differential equation.

Several forms of the formula underlie the Faddeev–LeVerrier algorithm for computing the characteristic polynomial, and explicit applications of the Cayley–Hamilton theorem. For example, starting from the following equation, which was proved above: