1. Databases
2. Proof database
3. Derivative of the determinant

Derivative of the determinant

There's two notions of the derivative we may need.

Theorem : Given the directional derivative of a matrix

\begin{eqnarray} \frac{\delta_B F[A]}{\delta A} &=& \lim_{\varepsilon \to 0} \frac{F[A + \varepsilon B] - F[A]}{\varepsilon} \end{eqnarray}

The directional derivative of the determinant is

\begin{eqnarray} \frac{\delta_B \det(A)}{\delta A} &=& \det(A) \mathrm{Tr}(A^{-1} B) \end{eqnarray}

Theorem : Given a function

\begin{eqnarray} A : I \subset \mathbb{R} &\to& M_{n \times n}\\ t &\mapsto& A(t) \end{eqnarray}