structural:scholar_edition
Derivative

The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from R n × n {\displaystyle \mathbf {R} ^{n\times n}} to R {\displaystyle \mathbf {R} }. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

where adj (A) {\displaystyle \operatorname {adj} (A)} denotes the adjugate of A {\displaystyle A}. In particular, if A {\displaystyle A} is invertible, we have

Expressed in terms of the entries of A {\displaystyle A}, these are

Yet another equivalent formulation is

using big O notation. The special case where A = I {\displaystyle A=I}, the identity matrix, yields

This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group SL n {\displaystyle \operatorname {SL} _{n}} is defined by the equation det A = 1 {\displaystyle \det A=1}. The above formula shows that its Lie algebra is the special linear Lie algebra s l n {\displaystyle {\mathfrak {sl}}_{n}} consisting of those matrices having trace zero.

Writing a 3 × 3 {\displaystyle 3\times 3} -matrix as A = {\displaystyle A={\begin{bmatrix}a&b&c\end{bmatrix}}} where a, b, c {\displaystyle a,b,c} are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two: