More simple testing data

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+#LyX 2.1 created this file. For more info see http://www.lyx.org/
+\lyxformat 474
+\begin_document
+\begin_header
+\textclass article
+\use_default_options true
+\maintain_unincluded_children false
+\language english
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman default
+\font_sans default
+\font_typewriter default
+\font_math auto
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100
+\font_tt_scale 100
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\use_hyperref false
+\papersize default
+\use_geometry false
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification true
+\use_refstyle 1
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\quotes_language english
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left[\begin{array}{ccc}
+\iota_{00} & \iota_{01} & \iota_{02}\\
+\iota_{10} & \iota_{11} & \iota_{12}\\
+\iota_{20} & \iota_{21} & \iota_{22}
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left[\begin{array}{ccc}
+{\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i1}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}}\\
+-{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i0}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}}\\
+-{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}} & {\displaystyle \sum_{i=1}^{n}}m_{i}\left(x_{i0}^{2}+x_{i1}^{2}\right)
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
+\left(x_{i1}^{2}+x_{i2}^{2}\right) & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
+-x_{i0}x_{i1} & \left(x_{i0}^{2}+x_{i2}^{2}\right) & -x_{i1}x_{i2}\\
+-x_{i0}x_{i2} & -x_{i1}x_{i2} & \left(x_{i0}^{2}+x_{i1}^{2}\right)
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
+\left|\mathbf{p}_{i}\right|^{2}-x_{i0}^{2} & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
+-x_{i0}x_{i1} & \left|\mathbf{p}_{i}\right|^{2}-x_{i1}^{2} & -x_{i1}x_{i2}\\
+-x_{i0}x_{i2} & -x_{i1}x_{i2} & \left|\mathbf{p}_{i}\right|^{2}-x_{i2}^{2}
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left[\begin{array}{ccc}
+\left|\mathbf{p}_{i}\right|^{2} & 0 & 0\\
+0 & \left|\mathbf{p}_{i}\right|^{2} & 0\\
+0 & 0 & \left|\mathbf{p}_{i}\right|^{2}
+\end{array}\right]-\left[\begin{array}{ccc}
+x_{i0}^{2} & x_{i0}x_{i1} & x_{i0}x_{i2}\\
+x_{i0}x_{i1} & x_{i1}^{2} & x_{i1}x_{i2}\\
+x_{i0}x_{i2} & x_{i1}x_{i2} & x_{i2}^{2}
+\end{array}\right]\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)-\boldsymbol{\mu}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)-\boldsymbol{\mu}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\mu}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{{\displaystyle \sum_{k=1}^{n}}m_{k}}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+M\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{\left({\displaystyle \sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+\frac{\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}^{\top}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\right)^{\top}-\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}\right)+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\frac{1}{M}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}^{\top}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)^{\top}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+--------------------------------------------------------------------------------
+----------------------------------------
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\frac{1}{M_{n}}\left|{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right|^{2}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n}=\left(\alpha_{n}-\frac{1}{M_{n}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}-\boldsymbol{\gamma}_{n}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n+1}=\left(\alpha_{n+1}-\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\right)\mathbf{I}+\left(\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}-\boldsymbol{\gamma}_{n+1}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n+1}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{M_{n}}{M_{n+1}}\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n}}\left[\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}-\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}+\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)\right]-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\alpha_{n+1}-\alpha_{n}=\sum_{i=1}^{n+1}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\alpha_{n+1}-\alpha_{n}=m_{n+1}\left|\mathbf{p}_{n+1}\right|^{2}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\boldsymbol{\beta}_{n}^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\boldsymbol{\beta}_{n}+m_{n+1}\mathbf{p}_{n+1}\right)+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+2m_{n+1}\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+m_{n+1}^{2}\frac{M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left(\frac{2M_{n}+m_{n+1}}{M_{n}+m_{n+1}}\right)\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+\frac{2m_{n+1}M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+\frac{m_{n+1}^{2}M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\frac{1}{M_{n}+m_{n+1}}\left[\boldsymbol{\beta}_{n}^{\top}\left(\left(2M_{n}+m_{n+1}\right)\boldsymbol{\beta}_{n}+2m_{n+1}M_{n}\mathbf{p}_{n+1}\right)+m_{n+1}^{2}M_{n}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

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