data/inertia.lyx

   1 #LyX 2.1 created this file. For more info see http://www.lyx.org/
   2 \lyxformat 474
   3 \begin_document
   4 \begin_header
   5 \textclass article
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   8 \language english
   9 \language_package default
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  28 \use_hyperref false
  29 \papersize default
  30 \use_geometry false
  31 \use_package amsmath 1
  32 \use_package amssymb 1
  33 \use_package cancel 1
  34 \use_package esint 1
  35 \use_package mathdots 1
  36 \use_package mathtools 1
  37 \use_package mhchem 1
  38 \use_package stackrel 1
  39 \use_package stmaryrd 1
  40 \use_package undertilde 1
  41 \cite_engine basic
  42 \cite_engine_type default
  43 \biblio_style plain
  44 \use_bibtopic false
  45 \use_indices false
  46 \paperorientation portrait
  47 \suppress_date false
  48 \justification true
  49 \use_refstyle 1
  50 \index Index
  51 \shortcut idx
  52 \color #008000
  53 \end_index
  54 \secnumdepth 3
  55 \tocdepth 3
  56 \paragraph_separation indent
  57 \paragraph_indentation default
  58 \quotes_language english
  59 \papercolumns 1
  60 \papersides 1
  61 \paperpagestyle default
  62 \tracking_changes false
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  64 \html_math_output 0
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  66 \html_be_strict false
  67 \end_header
  68
  69 \begin_body
  70
  71 \begin_layout Standard
  72 \begin_inset Formula
  73 \[
  74 \boldsymbol{\iota}=\left[\begin{array}{ccc}
  75 \iota_{00} & \iota_{01} & \iota_{02}\\
  76 \iota_{10} & \iota_{11} & \iota_{12}\\
  77 \iota_{20} & \iota_{21} & \iota_{22}
  78 \end{array}\right]
  79 \]
  80
  81 \end_inset
  82
  83
  84 \begin_inset Formula
  85 \[
  86 \boldsymbol{\iota}=\left[\begin{array}{ccc}
  87 {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i1}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}}\\
  88 -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i1}} & {\displaystyle \sum_{i=1}^{n}m_{i}\left(x_{i0}^{2}+x_{i2}^{2}\right)} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}}\\
  89 -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i0}x_{i2}} & -{\displaystyle \sum_{i=1}^{n}m_{i}x_{i1}x_{i2}} & {\displaystyle \sum_{i=1}^{n}}m_{i}\left(x_{i0}^{2}+x_{i1}^{2}\right)
  90 \end{array}\right]
  91 \]
  92
  93 \end_inset
  94
  95
  96 \begin_inset Formula
  97 \[
  98 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
  99 \left(x_{i1}^{2}+x_{i2}^{2}\right) & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
 100 -x_{i0}x_{i1} & \left(x_{i0}^{2}+x_{i2}^{2}\right) & -x_{i1}x_{i2}\\
 101 -x_{i0}x_{i2} & -x_{i1}x_{i2} & \left(x_{i0}^{2}+x_{i1}^{2}\right)
 102 \end{array}\right]
 103 \]
 104
 105 \end_inset
 106
 107
 108 \begin_inset Formula
 109 \[
 110 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left[\begin{array}{ccc}
 111 \left|\mathbf{p}_{i}\right|^{2}-x_{i0}^{2} & -x_{i0}x_{i1} & -x_{i0}x_{i2}\\
 112 -x_{i0}x_{i1} & \left|\mathbf{p}_{i}\right|^{2}-x_{i1}^{2} & -x_{i1}x_{i2}\\
 113 -x_{i0}x_{i2} & -x_{i1}x_{i2} & \left|\mathbf{p}_{i}\right|^{2}-x_{i2}^{2}
 114 \end{array}\right]
 115 \]
 116
 117 \end_inset
 118
 119
 120 \begin_inset Formula
 121 \[
 122 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left[\begin{array}{ccc}
 123 \left|\mathbf{p}_{i}\right|^{2} & 0 & 0\\
 124 0 & \left|\mathbf{p}_{i}\right|^{2} & 0\\
 125 0 & 0 & \left|\mathbf{p}_{i}\right|^{2}
 126 \end{array}\right]-\left[\begin{array}{ccc}
 127 x_{i0}^{2} & x_{i0}x_{i1} & x_{i0}x_{i2}\\
 128 x_{i0}x_{i1} & x_{i1}^{2} & x_{i1}x_{i2}\\
 129 x_{i0}x_{i2} & x_{i1}x_{i2} & x_{i2}^{2}
 130 \end{array}\right]\right)
 131 \]
 132
 133 \end_inset
 134
 135
 136 \begin_inset Formula
 137 \[
 138 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
 139 \]
 140
 141 \end_inset
 142
 143
 144 \begin_inset Formula
 145 \[
 146 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
 147 \]
 148
 149 \end_inset
 150
 151
 152 \begin_inset Formula
 153 \[
 154 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\mathbf{I}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}
 155 \]
 156
 157 \end_inset
 158
 159
 160 \begin_inset Formula
 161 \[
 162 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)^{\top}
 163 \]
 164
 165 \end_inset
 166
 167
 168 \begin_inset Formula
 169 \[
 170 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)
 171 \]
 172
 173 \end_inset
 174
 175
 176 \begin_inset Formula
 177 \[
 178 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)-\boldsymbol{\mu}^{\top}\left(\mathbf{p}_{i}-\boldsymbol{\mu}\right)\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)-\boldsymbol{\mu}\left(\mathbf{p}_{i}^{\top}-\boldsymbol{\mu}^{\top}\right)\right)
 179 \]
 180
 181 \end_inset
 182
 183
 184 \begin_inset Formula
 185 \[
 186 \boldsymbol{\iota}=\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\sum_{i=1}^{n}m_{i}\left(\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 187 \]
 188
 189 \end_inset
 190
 191
 192 \begin_inset Formula
 193 \[
 194 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 195 \]
 196
 197 \end_inset
 198
 199
 200 \begin_inset Formula
 201 \[
 202 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 203 \]
 204
 205 \end_inset
 206
 207
 208 \begin_inset Formula
 209 \[
 210 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\left(\sum_{i=1}^{n}m_{i}\right)\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 211 \]
 212
 213 \end_inset
 214
 215
 216 \begin_inset Formula
 217 \[
 218 \boldsymbol{\mu}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{{\displaystyle \sum_{k=1}^{n}}m_{k}}=\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}
 219 \]
 220
 221 \end_inset
 222
 223
 224 \end_layout
 225
 226 \begin_layout Standard
 227 \begin_inset Formula
 228 \[
 229 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\boldsymbol{\mu}-\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}^{\top}\mathbf{p}_{i}+M\boldsymbol{\mu}^{\top}\boldsymbol{\mu}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 230 \]
 231
 232 \end_inset
 233
 234
 235 \begin_inset Formula
 236 \[
 237 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\right)\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 238 \]
 239
 240 \end_inset
 241
 242
 243 \begin_inset Formula
 244 \[
 245 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{\left({\displaystyle \sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}-\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\mathbf{p}_{i}+\frac{\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)}{M}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 246 \]
 247
 248 \end_inset
 249
 250
 251 \begin_inset Formula
 252 \[
 253 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)\mathbf{I}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 254 \]
 255
 256 \end_inset
 257
 258
 259 \begin_inset Formula
 260 \[
 261 \boldsymbol{\iota}=\mathbf{I}^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 262 \]
 263
 264 \end_inset
 265
 266
 267 \begin_inset Formula
 268 \[
 269 \boldsymbol{\iota}=\mathbf{I}^{\top}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}\right)^{\top}-\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}^{\top}\right)\mathbf{p}_{i}\right)^{\top}\right)+\left(-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 270 \]
 271
 272 \end_inset
 273
 274
 275 \begin_inset Formula
 276 \[
 277 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\boldsymbol{\mu}^{\top}+\sum_{i=1}^{n}m_{i}\boldsymbol{\mu}\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\boldsymbol{\mu}\boldsymbol{\mu}^{\top}\right)
 278 \]
 279
 280 \end_inset
 281
 282
 283 \begin_inset Formula
 284 \[
 285 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-M\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)^{\top}\right)
 286 \]
 287
 288 \end_inset
 289
 290
 291 \begin_inset Formula
 292 \[
 293 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}+\sum_{i=1}^{n}m_{i}\left(\frac{{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}}{M}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}-\frac{1}{M}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}\right)
 294 \]
 295
 296 \end_inset
 297
 298
 299 \begin_inset Formula
 300 \[
 301 \boldsymbol{\iota}=\mathbf{I}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
 302 \]
 303
 304 \end_inset
 305
 306
 307 \begin_inset Formula
 308 \[
 309 \boldsymbol{\iota}^{\top}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\sum_{i=1}^{n}m_{i}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\mathbf{p}_{i}^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)^{\top}
 310 \]
 311
 312 \end_inset
 313
 314
 315 \begin_inset Formula
 316 \[
 317 \boldsymbol{\iota}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
 318 \]
 319
 320 \end_inset
 321
 322
 323 \end_layout
 324
 325 \begin_layout Standard
 326 --------------------------------------------------------------------------------
 327 ----------------------------------------
 328 \end_layout
 329
 330 \begin_layout Standard
 331 \begin_inset Formula
 332 \[
 333 \boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}^{\top}\mathbf{p}_{i}-\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
 334 \]
 335
 336 \end_inset
 337
 338
 339 \begin_inset Formula
 340 \[
 341 \boldsymbol{\iota}_{n}=\left(\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\frac{1}{M_{n}}\left|{\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right|^{2}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)\left({\displaystyle \sum_{j=1}^{n}}m_{j}\mathbf{p}_{j}\right)^{\top}-\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\mathbf{p}_{i}^{\top}\right)
 342 \]
 343
 344 \end_inset
 345
 346
 347 \begin_inset Formula
 348 \[
 349 \boldsymbol{\iota}_{n}=\left(\alpha_{n}-\frac{1}{M_{n}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}+\left(\frac{1}{M_{n}}\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}-\boldsymbol{\gamma}_{n}\right)
 350 \]
 351
 352 \end_inset
 353
 354
 355 \begin_inset Formula
 356 \[
 357 \boldsymbol{\iota}_{n+1}=\left(\alpha_{n+1}-\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\right)\mathbf{I}+\left(\frac{1}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}-\boldsymbol{\gamma}_{n+1}\right)
 358 \]
 359
 360 \end_inset
 361
 362
 363 \begin_inset Formula
 364 \[
 365 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n+1}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
 366 \]
 367
 368 \end_inset
 369
 370
 371 \begin_inset Formula
 372 \[
 373 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{M_{n}}{M_{n+1}}\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}\mathbf{I}-\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}\right)+\frac{1}{M_{n}}\left(\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\mathbf{I}-\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
 374 \]
 375
 376 \end_inset
 377
 378
 379 \begin_inset Formula
 380 \[
 381 \boldsymbol{\iota}_{n+1}-\boldsymbol{\iota}_{n}=\left(\alpha_{n+1}-\alpha_{n}\right)\mathbf{I}-\frac{1}{M_{n}}\left[\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}\right)\mathbf{I}-\left(\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}\boldsymbol{\beta}_{n+1}^{\top}+\boldsymbol{\beta}_{n}\boldsymbol{\beta}_{n}^{\top}\right)\right]-\left(\boldsymbol{\gamma}_{n+1}-\boldsymbol{\gamma}_{n}\right)
 382 \]
 383
 384 \end_inset
 385
 386
 387 \begin_inset Formula
 388 \[
 389 \alpha_{n+1}-\alpha_{n}=\sum_{i=1}^{n+1}m_{i}\left|\mathbf{p}_{i}\right|^{2}-\sum_{i=1}^{n}m_{i}\left|\mathbf{p}_{i}\right|^{2}
 390 \]
 391
 392 \end_inset
 393
 394
 395 \begin_inset Formula
 396 \[
 397 \alpha_{n+1}-\alpha_{n}=m_{n+1}\left|\mathbf{p}_{n+1}\right|^{2}
 398 \]
 399
 400 \end_inset
 401
 402
 403 \begin_inset Formula
 404 \[
 405 \frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n+1}^{\top}\boldsymbol{\beta}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}=\frac{M_{n}}{M_{n}+m_{n+1}}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
 406 \]
 407
 408 \end_inset
 409
 410
 411 \begin_inset Formula
 412 \[
 413 =\frac{M_{n}}{M_{n}+m_{n+1}}\left(\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}+m_{n+1}\mathbf{p}_{n+1}\right)+\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)^{\top}\left(\sum_{i=1}^{n}m_{i}\mathbf{p}_{i}\right)
 414 \]
 415
 416 \end_inset
 417
 418
 419 \begin_inset Formula
 420 \[
 421 =\frac{M_{n}}{M_{n}+m_{n+1}}\left(\boldsymbol{\beta}_{n}^{\top}+m_{n+1}\mathbf{p}_{n+1}^{\top}\right)\left(\boldsymbol{\beta}_{n}+m_{n+1}\mathbf{p}_{n+1}\right)+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
 422 \]
 423
 424 \end_inset
 425
 426
 427 \begin_inset Formula
 428 \[
 429 =\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+2m_{n+1}\frac{M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+m_{n+1}^{2}\frac{M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}+\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}
 430 \]
 431
 432 \end_inset
 433
 434
 435 \begin_inset Formula
 436 \[
 437 =\left(\frac{2M_{n}+m_{n+1}}{M_{n}+m_{n+1}}\right)\boldsymbol{\beta}_{n}^{\top}\boldsymbol{\beta}_{n}+\frac{2m_{n+1}M_{n}}{M_{n}+m_{n+1}}\boldsymbol{\beta}_{n}^{\top}\mathbf{p}_{n+1}+\frac{m_{n+1}^{2}M_{n}}{M_{n}+m_{n+1}}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}
 438 \]
 439
 440 \end_inset
 441
 442
 443 \begin_inset Formula
 444 \[
 445 =\frac{1}{M_{n}+m_{n+1}}\left[\boldsymbol{\beta}_{n}^{\top}\left(\left(2M_{n}+m_{n+1}\right)\boldsymbol{\beta}_{n}+2m_{n+1}M_{n}\mathbf{p}_{n+1}\right)+m_{n+1}^{2}M_{n}\mathbf{p}_{n+1}^{\top}\mathbf{p}_{n+1}\right]
 446 \]
 447
 448 \end_inset
 449
 450
 451 \end_layout
 452
 453 \end_body
 454 \end_document