derivative of matrix

Suppose $I$ is an open set of $ℝ$, 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 $A$,

   ${\left({\displaystyle \frac{dA}{dt}}\right)}^T$ $=$ ${\displaystyle \frac{d}{dt}}\left(A^T\right),$

   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 $A(t)$,

   $\mathrm{tr}({\displaystyle \frac{dA}{dt}})$ $=$ ${\displaystyle \frac{d}{dt}}\mathrm{tr}(A),$

   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	2013-03-22 15:00:28
Last modified on	2013-03-22 15:00:28
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	10
Author	matte (1858)
Entry type	Definition
Classification	msc 15-01
Related topic	NthDerivativeOfADeterminant