Editing Determinant (section)

=== Derivative ===
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a [[polynomial]] function from <math>\mathbf R^{n \times n}</math> to <math>\mathbf R</math>. In particular, it is everywhere [[differentiable]]. Its derivative can be expressed using [[Jacobi's formula]]:<ref>{{harvnb|Horn|Johnson|2018|loc=§&nbsp;0.8.10}}</ref>

:<math>\frac{d \det(A)}{d \alpha} = \operatorname{tr}\left(\operatorname{adj}(A) \frac{d A}{d \alpha}\right).</math>

where <math>\operatorname{adj}(A)</math> denotes the [[adjugate]] of <math>A</math>. In particular, if <math>A</math> is invertible, we have

:<math>\frac{d \det(A)}{d \alpha} = \det(A) \operatorname{tr}\left(A^{-1} \frac{d A}{d \alpha}\right).</math>

Expressed in terms of the entries of <math>A</math>, these are

: <math> \frac{\partial \det(A)}{\partial A_{ij}}= \operatorname{adj}(A)_{ji} = \det(A)\left(A^{-1}\right)_{ji}.</math>

Yet another equivalent formulation is

:<math>\det(A + \epsilon X) - \det(A) = \operatorname{tr}(\operatorname{adj}(A) X) \epsilon + O\left(\epsilon^2\right) = \det(A) \operatorname{tr}\left(A^{-1} X\right) \epsilon + O\left(\epsilon^2\right)</math>,

using [[big O notation]]. The special case where <math>A = I</math>, the identity matrix, yields

:<math>\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O\left(\epsilon^2\right).</math>

This identity is used in describing [[Lie algebra]]s associated to certain matrix [[Lie group]]s. For example, the special linear group <math>\operatorname{SL}_n</math> is defined by the equation <math>\det A = 1</math>. The above formula shows that its Lie algebra is the [[special linear Lie algebra]] <math>\mathfrak{sl}_n</math> consisting of those matrices having trace zero.

Writing a <math>3 \times 3</math> matrix as <math>A = \begin{bmatrix}a & b & c\end{bmatrix}</math> where <math>a, b,c</math> 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:

: <math>\begin{align}
  \nabla_\mathbf{a}\det(A) &= \mathbf{b} \times \mathbf{c} \\
  \nabla_\mathbf{b}\det(A) &= \mathbf{c} \times \mathbf{a} \\
  \nabla_\mathbf{c}\det(A) &= \mathbf{a} \times \mathbf{b}.
\end{align}</math>