exercises.tex

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\title{Exercises of State Estimation for Robotics}

% put your name here
\author{Xiang Gao}

\begin{document}

\maketitle

\section{Chapter 2}
\subsection{ex 2.1}
\begin{equation}
{\mathbf{u}^T}\mathbf{v} = \sum\limits_{i = 1}^n {{u_i}{v_i}}  = \mathrm{tr}\left( {\mathbf{v}{\mathbf{u}^T}} \right)
\end{equation}

\subsection{ex 2.2}
\begin{equation}
\begin{split}
H\left( {\mathbf{x},\mathbf{y}} \right) &=  - \iint {p\left( {\mathbf{x},\mathbf{y}} \right)\ln }p\left( {\mathbf{x},\mathbf{y}} \right)\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y} \\ 
&=  - \iint {p\left( \mathbf{x} \right)p\left( \mathbf{y} \right)\left( {\ln p\left( \mathbf{x} \right) + \ln p\left( \mathbf{y} \right)} \right)\mathrm{d}\mathbf{x}\mathrm{d}\mathbf{y}} \\ 
&=  - \int {p\left( \mathbf{x} \right)\ln p\left( \mathbf{x} \right)\underbrace {\left( {\int {p\left( \mathbf{y} \right)\mathrm{d}\mathbf{y}} } \right)}_1\mathrm{d}\mathbf{x} - \int {p\left( \mathbf{y} \right)\ln p\left( \mathbf{y} \right)\underbrace {\left( {\int {p\left( \mathbf{x} \right)\mathrm{d}\mathbf{x}} } \right)}_1\mathrm{d}\mathbf{y}} }  \\ 
&= H\left( \mathbf{x} \right) + H\left( \mathbf{y} \right) \\ 
\end{split}
\end{equation}

\subsection{ex 2.3}
\begin{equation}
\begin{split}
E\left( {\mathbf{x}{\mathbf{x}^T}} \right) &= E\left( {\left( {\mathbf{x} - \boldsymbol{\mu}  + \boldsymbol{\mu} } \right){{\left( {\mathbf{x} - \boldsymbol{\mu}  + \boldsymbol{\mu} } \right)}^T}} \right) \hfill \\
&= E\left( {\underbrace {\left( {\mathbf{x} - \boldsymbol{\mu} } \right){{\left( {\mathbf{x} - \boldsymbol{\mu} } \right)}^T}}_{\boldsymbol{\Sigma}}  + \underbrace {\boldsymbol{\mu} {{\left( {\mathbf{x} - \boldsymbol{\mu} } \right)}^T}}_{\boldsymbol{\mu} {\mathbf{0}^T}} + \underbrace {\left( {\mathbf{x} - \boldsymbol{\mu} } \right){\boldsymbol{\mu} ^T}}_{\mathbf{0}{\boldsymbol{\mu} ^T}} + \boldsymbol{\mu} {\boldsymbol{\mu} ^T}} \right) \hfill \\
&= \boldsymbol{\Sigma}  + \boldsymbol{\mu} {\boldsymbol{\mu} ^T} \hfill \\ 
\end{split}
\end{equation}


\end{document}