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.^[1] If A is a differentiable map from the real numbers to n × n matrices,

$\frac{d}{dt} \det A(t) = \mathrm{tr} \left (\mathrm{adj}(A(t)) \, \frac{dA(t)}{dt}\right )~$

where tr(X) is the trace of the matrix X. Equivalently, if dA stands for the differential of A, the formula is

$d \det (A) = \mathrm{tr} (\mathrm{adj}(A) \, dA).$

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

Derivation

We first prove a preliminary lemma:

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

$\sum_i \sum_j A_{ij} B_{ij} = \mathrm{tr} (A^{\rm T} B).$

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

$(AB)_{jk} = \sum_i A_{ji} B_{ik}.\,$

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

$(A^{\rm T} B)_{jk} = \sum_i A_{ij} B_{ik}.$

The result follows by taking the trace of both sides:

$\mathrm{tr} (A^{\rm T} B) = \sum_j (A^{\rm T} B)_{jj} = \sum_j \sum_i A_{ij} B_{ij} = \sum_i \sum_j A_{ij} B_{ij}.\ \square$

Theorem. (Jacobi's formula) For any differentiable map A from the real numbers to n × n matrices,

$d \det (A) = \mathrm{tr} (\mathrm{adj}(A) \, dA).$

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

$\det(A) = \sum_j A_{ij} \mathrm{adj}^{\rm T} (A)_{ij}.$

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:

$\det(A) = F\,(A_{11}, A_{12}, \ldots , A_{21}, A_{22}, \ldots , A_{nn})$

so that, by the chain rule, its differential is

$d \det(A) = \sum_i \sum_j {\partial F \over \partial A_{ij}} \,dA_{ij}.$

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

To find ∂F/∂A_ij 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 ∂ / ∂A_ij:

${\partial \det(A) \over \partial A_{ij}} = {\partial \sum_k A_{ik} \mathrm{adj}^{\rm T}(A)_{ik} \over \partial A_{ij}} = \sum_k {\partial (A_{ik} \mathrm{adj}^{\rm T}(A)_{ik}) \over \partial A_{ij}}$

Thus, by the product rule,

${\partial \det(A) \over \partial A_{ij}} = \sum_k {\partial A_{ik} \over \partial A_{ij}} \mathrm{adj}^{\rm T}(A)_{ik} + \sum_k A_{ik} {\partial \, \mathrm{adj}^{\rm T}(A)_{ik} \over \partial A_{ij}}.$

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,

${\partial \, \mathrm{adj}^{\rm T}(A)_{ik} \over \partial A_{ij}} = 0,$

so

${\partial \det(A) \over \partial A_{ij}} = \sum_k \mathrm{adj}^{\rm T}(A)_{ik} {\partial A_{ik} \over \partial A_{ij}}.$

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

${\partial A_{ik} \over \partial A_{ij}} = \delta_{jk},$

where δ is the Kronecker delta, so

${\partial \det(A) \over \partial A_{ij}} = \sum_k \mathrm{adj}^{\rm T}(A)_{ik} \delta_{jk} = \mathrm{adj}^{\rm T}(A)_{ij}.$

Therefore,

$d(\det(A)) = \sum_i \sum_j \mathrm{adj}^{\rm T}(A)_{ij} \,d A_{ij},$

and applying the Lemma yields

$d(\det(A)) = \mathrm{tr}(\mathrm{adj}(A) \,dA).\ \square$

Corollary

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

$\frac{d}{dt} \det A(t) = (\det A(t)) \, \mathrm{tr} \left(A(t)^{-1} \, \frac{d}{dt} A(t)\right)$

which can be alternatively written as

$\frac{d}{dt} \log \det A(t) = \mathrm{tr} \left(A(t)^{-1} \, \frac{d}{dt} A(t)\right).$

Furthermore, taking A(t) = exp(tB) in the first equation, we obtain

$\frac{d}{dt} \det e^{tB} =\mathrm{tr} \left(B\right) \det e^{tB} ,$

solved by

$\det e^{tB} = e^{\mathrm{tr} \left(tB\right)},$

a useful relation connecting the trace to the determinant of the associated matrix exponential.

Notes

1. Magnus & Neudecker (1999), Part Three, Section 8.3

References

• Magnus, Jan R.; Neudecker, Heinz (1999), Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley, ISBN 0-471-98633-X
• Bellmann, Richard (1987), Introduction to Matrix Analysis, SIAM, ISBN 0898713994