# derivative of matrix

Suppose $I$ is an open set of $\mathbbmss{R}$, 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. 1.

For any $n\times m$ matrix $A$,

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

where ${}^{T}$ is the matrix transpose.

2. 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. 3.

When $A(t)$ is invertible,

 $\frac{d}{dt}(A^{-1})=-A^{-1}\frac{dA}{dt}A^{-1}.$
4. 4.

For a square matrix $A(t)$,

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

where $\operatorname{tr}$ is the matrix trace.

5. 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 DerivativeOfMatrix 2013-03-22 15:00:28 2013-03-22 15:00:28 matte (1858) matte (1858) 10 matte (1858) Definition msc 15-01 NthDerivativeOfADeterminant