15-01-DerivativeOfMatrix.tex

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\pmtitle{derivative of matrix}
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\pmtype{Definition}
\pmcomment{trigger rebuild}
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\pmrelated{NthDerivativeOfADeterminant}

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\begin{document}
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 $\frac{dA}{dt}$. 

\subsubsection*{Properties}
In the below we assume that all matrices are dependent on a parameter $t$
and the matrices are differentiable with respect to $t$. 
\begin{enumerate}
\item For any $n\times m$ matrix $A$,
\begin{eqnarray*}
   \left(\frac{dA}{dt}\right)^T &=& \frac{d}{dt}\left(A^T\right), 
\end{eqnarray*}
where $^T$ is the matrix transpose.
\item If $A(t),B(t)$ are matrices such that $AB$ is defined, then 
$$
   \frac{d}{dt}(AB) = \frac{dA}{dt} B + A\frac{dB}{dt}.
$$
\item When $A(t)$ is invertible,
$$
   \frac{d}{dt}(A^{-1}) = -A^{-1} \frac{dA}{dt} A^{-1}.
$$
\item For a square matrix $A(t)$, 
\begin{eqnarray*}
   \operatorname{tr}(\frac{dA}{dt}) &=& \frac{d}{dt}\operatorname{tr}(A),
\end{eqnarray*}
where $\operatorname{tr}$ is the matrix trace.
\item If $A(t),B(t)$ are $n\times m$ matrices and $A\circ B$ is the
Hadamard product of $A$ and $B$, then 
$$
   \frac{d}{dt}(A\circ B) = \frac{dA}{dt}\circ  B + A\circ \frac{dB}{dt}.
$$
%\operatorname{exp} is the matrix exponential.
\end{enumerate}
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\end{document}