derivative of matrix
Suppose $I$ is an open set of $\mathbb{R}$, and for each $t\in I$, $A(t)$ is an $n\times m$ matrix. If each element in $A(t)$ is a differentiable function of $t$, we say that $A$ is a differentiable^{}, and define the derivative of $A$ componentwise. This derivative we shall write as $\frac{d}{dt}A$ or $\frac{dA}{dt}$.
Properties
In the below we assume that all matrices are dependent on a parameter $t$ and the matrices are differentiable with respect to $t$.

1.
For any $n\times m$ matrix $A$,
${\left({\displaystyle \frac{dA}{dt}}\right)}^{T}$ $=$ $\frac{d}{dt}}\left({A}^{T}\right),$ where ${}^{T}$ is the matrix transpose.

2.
If $A(t),B(t)$ are matrices such that $AB$ is defined, then
$$\frac{d}{dt}(AB)=\frac{dA}{dt}B+A\frac{dB}{dt}.$$ 
3.
When $A(t)$ is invertible^{},
$$\frac{d}{dt}({A}^{1})={A}^{1}\frac{dA}{dt}{A}^{1}.$$ 
4.
For a square matrix^{} $A(t)$,
$\mathrm{tr}({\displaystyle \frac{dA}{dt}})$ $=$ $\frac{d}{dt}}\mathrm{tr}(A),$ where $\mathrm{tr}$ is the matrix trace.

5.
If $A(t),B(t)$ are $n\times m$ matrices and $A\circ B$ is the Hadamard product of $A$ and $B$, then
$$\frac{d}{dt}(A\circ B)=\frac{dA}{dt}\circ B+A\circ \frac{dB}{dt}.$$
Title  derivative of matrix 

Canonical name  DerivativeOfMatrix 
Date of creation  20130322 15:00:28 
Last modified on  20130322 15:00:28 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  10 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 1501 
Related topic  NthDerivativeOfADeterminant 