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maint: Use Octave coding conventions for cuddling parentheses in doc directory * OctaveFAQ.texi, basics.txi, container.txi, contrib.txi, diagperm.txi, diffeq.txi, dynamic.txi, errors.txi, eval.txi, expr.txi, func.txi, geometry.txi, interp.txi, intro.txi, numbers.txi, oop.txi, plot.txi, poly.txi, quad.txi, set.txi, sparse.txi, stmt.txi, testfun.txi, vectorize.txi, refcard.tex: Use Octave coding conventions for cuddling parentheses.
author Rik <octave@nomad.inbox5.com>
date Mon, 09 Jul 2012 17:00:46 -0700
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1 @c Copyright (C) 2009-2012 Jaroslav Hajek
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2 @c
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3 @c This file is part of Octave.
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4 @c
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5 @c Octave is free software; you can redistribute it and/or modify it
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6 @c under the terms of the GNU General Public License as published by the
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7 @c Free Software Foundation; either version 3 of the License, or (at
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8 @c your option) any later version.
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9 @c
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10 @c Octave is distributed in the hope that it will be useful, but WITHOUT
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11 @c ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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12 @c FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
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13 @c for more details.
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14 @c
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15 @c You should have received a copy of the GNU General Public License
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16 @c along with Octave; see the file COPYING. If not, see
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17 @c <http://www.gnu.org/licenses/>.
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18
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19 @node Diagonal and Permutation Matrices
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20 @chapter Diagonal and Permutation Matrices
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21
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22 @menu
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23 * Basic Usage:: Creation and Manipulation of Diagonal and Permutation Matrices
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24 * Matrix Algebra:: Linear Algebra with Diagonal and Permutation Matrices
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25 * Function Support:: Functions That Are Aware of These Matrices
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26 * Example Code:: Some Examples of Usage
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27 * Zeros Treatment:: The Differences in Treatment of Zero Elements
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28 @end menu
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29
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30 @node Basic Usage
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31 @section Creating and Manipulating Diagonal and Permutation Matrices
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32
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33 A diagonal matrix is defined as a matrix that has zero entries outside the main
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34 diagonal; that is,
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35 @tex
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36 $D_{ij} = 0$ if $i \neq j$
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37 @end tex
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38 @ifnottex
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39 @code{D(i,j) == 0} if @code{i != j}.
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40 @end ifnottex
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41 Most often, square diagonal matrices are considered; however, the definition can
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42 equally be applied to non-square matrices, in which case we usually speak of a
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43 rectangular diagonal matrix.
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44
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45 A permutation matrix is defined as a square matrix that has a single element
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46 equal to unity in each row and each column; all other elements are zero. That
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47 is, there exists a permutation (vector)
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48 @tex
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49 $p$ such that $P_{ij}=1$ if $j = p_i$ and
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50 $P_{ij}=0$ otherwise.
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51 @end tex
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52 @ifnottex
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53 @code{p} such that @code{P(i,j) == 1} if @code{j == p(i)} and
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54 @code{P(i,j) == 0} otherwise.
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55 @end ifnottex
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56
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57 Octave provides special treatment of real and complex rectangular diagonal
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58 matrices, as well as permutation matrices. They are stored as special objects,
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59 using efficient storage and algorithms, facilitating writing both readable and
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60 efficient matrix algebra expressions in the Octave language.
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61
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62 @menu
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63 * Creating Diagonal Matrices::
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64 * Creating Permutation Matrices::
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65 * Explicit and Implicit Conversions::
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66 @end menu
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67
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68 @node Creating Diagonal Matrices
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69 @subsection Creating Diagonal Matrices
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70
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71 The most common and easiest way to create a diagonal matrix is using the
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72 built-in function @dfn{diag}. The expression @code{diag (v)}, with @var{v} a
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73 vector, will create a square diagonal matrix with elements on the main diagonal
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74 given by the elements of @var{v}, and size equal to the length of @var{v}.
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75 @code{diag (v, m, n)} can be used to construct a rectangular diagonal matrix.
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76 The result of these expressions will be a special diagonal matrix object, rather
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77 than a general matrix object.
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78
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79 Diagonal matrix with unit elements can be created using @dfn{eye}.
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80 Some other built-in functions can also return diagonal matrices. Examples
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81 include
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82 @dfn{balance} or @dfn{inv}.
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83
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84 Example:
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85
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86 @example
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87 diag (1:4)
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88 @result{}
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89 Diagonal Matrix
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90
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91 1 0 0 0
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92 0 2 0 0
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93 0 0 3 0
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94 0 0 0 4
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95
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96 diag (1:3,5,3)
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97
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98 @result{}
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99 Diagonal Matrix
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100
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101 1 0 0
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102 0 2 0
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103 0 0 3
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104 0 0 0
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105 0 0 0
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106 @end example
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107
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108 @node Creating Permutation Matrices
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109 @subsection Creating Permutation Matrices
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110
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111 For creating permutation matrices, Octave does not introduce a new function, but
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112 rather overrides an existing syntax: permutation matrices can be conveniently
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113 created by indexing an identity matrix by permutation vectors.
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114 That is, if @var{q} is a permutation vector of length @var{n}, the expression
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115
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116 @example
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117 P = eye (n) (:, q);
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118 @end example
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119
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120 @noindent
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121 will create a permutation matrix - a special matrix object.
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122
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123 @example
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124 eye (n) (q, :)
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125 @end example
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126
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127 @noindent
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128 will also work (and create a row permutation matrix), as well as
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129
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130 @example
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131 eye (n) (q1, q2).
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132 @end example
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133
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134 For example:
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135
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136 @example
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137 @group
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138 eye (4) ([1,3,2,4],:)
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139 @result{}
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140 Permutation Matrix
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141
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142 1 0 0 0
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143 0 0 1 0
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144 0 1 0 0
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145 0 0 0 1
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146
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147 eye (4) (:,[1,3,2,4])
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148 @result{}
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149 Permutation Matrix
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150
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151 1 0 0 0
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152 0 0 1 0
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153 0 1 0 0
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154 0 0 0 1
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155 @end group
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156 @end example
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157
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158 Mathematically, an identity matrix is both diagonal and permutation matrix.
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159 In Octave, @code{eye (n)} returns a diagonal matrix, because a matrix
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160 can only have one class. You can convert this diagonal matrix to a permutation
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161 matrix by indexing it by an identity permutation, as shown below.
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162 This is a special property of the identity matrix; indexing other diagonal
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163 matrices generally produces a full matrix.
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164
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165 @example
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166 @group
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167 eye (3)
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168 @result{}
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169 Diagonal Matrix
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170
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171 1 0 0
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172 0 1 0
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173 0 0 1
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174
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175 eye(3)(1:3,:)
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176 @result{}
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177 Permutation Matrix
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178
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179 1 0 0
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180 0 1 0
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181 0 0 1
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182 @end group
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183 @end example
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184
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185 Some other built-in functions can also return permutation matrices. Examples
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186 include
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187 @dfn{inv} or @dfn{lu}.
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188
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189 @node Explicit and Implicit Conversions
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190 @subsection Explicit and Implicit Conversions
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191
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192 The diagonal and permutation matrices are special objects in their own right. A
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193 number of operations and built-in functions are defined for these matrices to
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194 use special, more efficient code than would be used for a full matrix in the
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195 same place. Examples are given in further sections.
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196
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197 To facilitate smooth mixing with full matrices, backward compatibility, and
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198 compatibility with @sc{matlab}, the diagonal and permutation matrices should
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199 allow any operation that works on full matrices, and will either treat it
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200 specially, or implicitly convert themselves to full matrices.
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201
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202 Instances include matrix indexing, except for extracting a single element or
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203 a leading submatrix, indexed assignment, or applying most mapper functions,
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204 such as @dfn{exp}.
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205
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206 An explicit conversion to a full matrix can be requested using the built-in
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207 function @dfn{full}. It should also be noted that the diagonal and permutation
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208 matrix objects will cache the result of the conversion after it is first
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209 requested (explicitly or implicitly), so that subsequent conversions will
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210 be very cheap.
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211
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212 @node Matrix Algebra
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213 @section Linear Algebra with Diagonal and Permutation Matrices
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214
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215 As has been already said, diagonal and permutation matrices make it
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216 possible to use efficient algorithms while preserving natural linear
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217 algebra syntax. This section describes in detail the operations that
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218 are treated specially when performed on these special matrix objects.
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219
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220 @menu
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221 * Expressions Involving Diagonal Matrices::
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222 * Expressions Involving Permutation Matrices::
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223 @end menu
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224
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225 @node Expressions Involving Diagonal Matrices
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226 @subsection Expressions Involving Diagonal Matrices
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227
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228 Assume @var{D} is a diagonal matrix. If @var{M} is a full matrix,
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229 then @code{D*M} will scale the rows of @var{M}. That means,
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230 if @code{S = D*M}, then for each pair of indices
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231 i,j it holds
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232 @tex
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233 $$S_{ij} = D_{ii} M_{ij}$$
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234 @end tex
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235 @ifnottex
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236
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237 @example
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238 S(i,j) = D(i,i) * M(i,j).
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239 @end example
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240
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241 @end ifnottex
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242 Similarly, @code{M*D} will do a column scaling.
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243
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244 The matrix @var{D} may also be rectangular, m-by-n where @code{m != n}.
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245 If @code{m < n}, then the expression @code{D*M} is equivalent to
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246
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247 @example
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248 D(:,1:m) * M(1:m,:),
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249 @end example
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250
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251 @noindent
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252 i.e., trailing @code{n-m} rows of @var{M} are ignored. If @code{m > n},
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253 then @code{D*M} is equivalent to
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254
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255 @example
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256 [D(1:n,n) * M; zeros(m-n, columns (M))],
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257 @end example
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258
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259 @noindent
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260 i.e., null rows are appended to the result.
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261 The situation for right-multiplication @code{M*D} is analogous.
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262
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263 The expressions @code{D \ M} and @code{M / D} perform inverse scaling.
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264 They are equivalent to solving a diagonal (or rectangular diagonal)
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265 in a least-squares minimum-norm sense. In exact arithmetic, this is
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266 equivalent to multiplying by a pseudoinverse. The pseudoinverse of
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267 a rectangular diagonal matrix is again a rectangular diagonal matrix
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268 with swapped dimensions, where each nonzero diagonal element is replaced
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269 by its reciprocal.
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270 The matrix division algorithms do, in fact, use division rather than
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271 multiplication by reciprocals for better numerical accuracy; otherwise, they
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272 honor the above definition. Note that a diagonal matrix is never truncated due
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273 to ill-conditioning; otherwise, it would not be much useful for scaling. This
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274 is typically consistent with linear algebra needs. A full matrix that only
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275 happens to be diagonal (an is thus not a special object) is of course treated
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276 normally.
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277
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278 Multiplication and division by diagonal matrices works efficiently also when
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279 combined with sparse matrices, i.e., @code{D*S}, where @var{D} is a diagonal
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280 matrix and @var{S} is a sparse matrix scales the rows of the sparse matrix and
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281 returns a sparse matrix. The expressions @code{S*D}, @code{D\S}, @code{S/D}
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282 work analogically.
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283
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284 If @var{D1} and @var{D2} are both diagonal matrices, then the expressions
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285
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286 @example
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287 @group
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288 D1 + D2
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289 D1 - D2
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290 D1 * D2
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291 D1 / D2
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292 D1 \ D2
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293 @end group
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294 @end example
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295
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296 @noindent
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297 again produce diagonal matrices, provided that normal
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298 dimension matching rules are obeyed. The relations used are same as described
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299 above.
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300
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301 Also, a diagonal matrix @var{D} can be multiplied or divided by a scalar, or
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302 raised to a scalar power if it is square, producing diagonal matrix result in
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303 all cases.
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304
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305 A diagonal matrix can also be transposed or conjugate-transposed, giving the
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306 expected result. Extracting a leading submatrix of a diagonal matrix, i.e.,
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307 @code{D(1:m,1:n)}, will produce a diagonal matrix, other indexing expressions
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308 will implicitly convert to full matrix.
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309
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310 Adding a diagonal matrix to a full matrix only operates on the diagonal
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311 elements. Thus,
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312
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313 @example
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314 A = A + eps * eye (n)
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315 @end example
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316
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317 @noindent
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318 is an efficient method of augmenting the diagonal of a matrix. Subtraction
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319 works analogically.
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320
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321 When involved in expressions with other element-by-element operators, @code{.*},
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322 @code{./}, @code{.\} or @code{.^}, an implicit conversion to full matrix will
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323 take place. This is not always strictly necessary but chosen to facilitate
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324 better consistency with @sc{matlab}.
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325
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326 @node Expressions Involving Permutation Matrices
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327 @subsection Expressions Involving Permutation Matrices
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328
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329 If @var{P} is a permutation matrix and @var{M} a matrix, the expression
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330 @code{P*M} will permute the rows of @var{M}. Similarly, @code{M*P} will
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331 yield a column permutation.
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332 Matrix division @code{P\M} and @code{M/P} can be used to do inverse permutation.
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333
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334 The previously described syntax for creating permutation matrices can actually
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335 help an user to understand the connection between a permutation matrix and
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336 a permuting vector. Namely, the following holds, where @code{I = eye (n)}
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337 is an identity matrix:
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338
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339 @example
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340 I(p,:) * M = (I*M) (p,:) = M(p,:)
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341 @end example
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342
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343 Similarly,
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344
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345 @example
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346 M * I(:,p) = (M*I) (:,p) = M(:,p)
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347 @end example
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348
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349 The expressions @code{I(p,:)} and @code{I(:,p)} are permutation matrices.
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350
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351 A permutation matrix can be transposed (or conjugate-transposed, which is the
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352 same, because a permutation matrix is never complex), inverting the
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353 permutation, or equivalently, turning a row-permutation matrix into a
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354 column-permutation one. For permutation matrices, transpose is equivalent to
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355 inversion, thus @code{P\M} is equivalent to @code{P'*M}. Transpose of a
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356 permutation matrix (or inverse) is a constant-time operation, flipping only a
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357 flag internally, and thus the choice between the two above equivalent
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358 expressions for inverse permuting is completely up to the user's taste.
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359
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360 Multiplication and division by permutation matrices works efficiently also when
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361 combined with sparse matrices, i.e., @code{P*S}, where @var{P} is a permutation
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362 matrix and @var{S} is a sparse matrix permutes the rows of the sparse matrix and
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363 returns a sparse matrix. The expressions @code{S*P}, @code{P\S}, @code{S/P}
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364 work analogically.
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365
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366 Two permutation matrices can be multiplied or divided (if their sizes match),
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367 performing a composition of permutations. Also a permutation matrix can be
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368 indexed by a permutation vector (or two vectors), giving again a permutation
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369 matrix. Any other operations do not generally yield a permutation matrix and
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370 will thus trigger the implicit conversion.
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371
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372 @node Function Support
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373 @section Functions That Are Aware of These Matrices
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374
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375 This section lists the built-in functions that are aware of diagonal and
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376 permutation matrices on input, or can return them as output. Passed to other
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377 functions, these matrices will in general trigger an implicit conversion.
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378 (Of course, user-defined dynamically linked functions may also work with
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379 diagonal or permutation matrices).
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380
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381 @menu
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382 * Diagonal Matrix Functions::
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383 * Permutation Matrix Functions::
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384 @end menu
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385
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386 @node Diagonal Matrix Functions
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387 @subsection Diagonal Matrix Functions
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388
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389 @dfn{inv} and @dfn{pinv} can be applied to a diagonal matrix, yielding again
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390 a diagonal matrix. @dfn{det} will use an efficient straightforward calculation
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391 when given a diagonal matrix, as well as @dfn{cond}.
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392 The following mapper functions can be applied to a diagonal matrix
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393 without converting it to a full one:
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394 @dfn{abs}, @dfn{real}, @dfn{imag}, @dfn{conj}, @dfn{sqrt}.
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395 A diagonal matrix can also be returned from the @dfn{balance}
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396 and @dfn{svd} functions.
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397 The @dfn{sparse} function will convert a diagonal matrix efficiently to a
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398 sparse matrix.
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399
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400 @node Permutation Matrix Functions
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401 @subsection Permutation Matrix Functions
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402
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403 @dfn{inv} and @dfn{pinv} will invert a permutation matrix, preserving its
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404 specialness. @dfn{det} can be applied to a permutation matrix, efficiently
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405 calculating the sign of the permutation (which is equal to the determinant).
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406
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407 A permutation matrix can also be returned from the built-in functions
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408 @dfn{lu} and @dfn{qr}, if a pivoted factorization is requested.
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409
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410 The @dfn{sparse} function will convert a permutation matrix efficiently to a
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411 sparse matrix.
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412 The @dfn{find} function will also work efficiently with a permutation matrix,
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413 making it possible to conveniently obtain the permutation indices.
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414
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415 @node Example Code
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416 @section Some Examples of Usage
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417
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418 The following can be used to solve a linear system @code{A*x = b}
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419 using the pivoted LU@tie{}factorization:
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420
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421 @example
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422 @group
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423 [L, U, P] = lu (A); ## now L*U = P*A
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424 x = U \ L \ P*b;
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425 @end group
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426 @end example
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427
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428 @noindent
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429 This is one way to normalize columns of a matrix @var{X} to unit norm:
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430
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431 @example
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432 @group
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433 s = norm (X, "columns");
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434 X /= diag (s);
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435 @end group
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436 @end example
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437
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438 @noindent
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439 The same can also be accomplished with broadcasting
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440 (@pxref{Broadcasting}):
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441
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442 @example
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443 @group
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444 s = norm (X, "columns");
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445 X ./= s;
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446 @end group
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447 @end example
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448
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449 @noindent
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450 The following expression is a way to efficiently calculate the sign of a
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451 permutation, given by a permutation vector @var{p}. It will also work
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452 in earlier versions of Octave, but slowly.
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453
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454 @example
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455 det (eye (length (p))(p, :))
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456 @end example
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457
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458 @noindent
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459 Finally, here's how you solve a linear system @code{A*x = b}
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460 with Tikhonov regularization (ridge regression) using SVD (a skeleton only):
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461
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462 @example
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463 @group
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464 m = rows (A); n = columns (A);
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465 [U, S, V] = svd (A);
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466 ## determine the regularization factor alpha
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467 ## alpha = @dots{}
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468 ## transform to orthogonal basis
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469 b = U'*b;
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470 ## Use the standard formula, replacing A with S.
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471 ## S is diagonal, so the following will be very fast and accurate.
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472 x = (S'*S + alpha^2 * eye (n)) \ (S' * b);
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473 ## transform to solution basis
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474 x = V*x;
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475 @end group
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476 @end example
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477
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478
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479 @node Zeros Treatment
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480 @section The Differences in Treatment of Zero Elements
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481
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482 Making diagonal and permutation matrices special matrix objects in their own
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483 right and the consequent usage of smarter algorithms for certain operations
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484 implies, as a side effect, small differences in treating zeros.
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485 The contents of this section applies also to sparse matrices, discussed in
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486 the following chapter.
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487
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488 The IEEE standard defines the result of the expressions @code{0*Inf} and
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489 @code{0*NaN} as @code{NaN}, as it has been generally agreed that this is the
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490 best compromise.
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491 Numerical software dealing with structured and sparse matrices (including
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492 Octave) however, almost always makes a distinction between a "numerical zero"
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493 and an "assumed zero".
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494 A "numerical zero" is a zero value occurring in a place where any floating-point
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495 value could occur. It is normally stored somewhere in memory as an explicit
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496 value.
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497 An "assumed zero", on the contrary, is a zero matrix element implied by the
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498 matrix structure (diagonal, triangular) or a sparsity pattern; its value is
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499 usually not stored explicitly anywhere, but is implied by the underlying
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500 data structure.
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501
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502 The primary distinction is that an assumed zero, when multiplied
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503 by any number, or divided by any nonzero number,
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504 yields *always* a zero, even when, e.g., multiplied by @code{Inf}
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505 or divided by @code{NaN}.
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506 The reason for this behavior is that the numerical multiplication is not
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507 actually performed anywhere by the underlying algorithm; the result is
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508 just assumed to be zero. Equivalently, one can say that the part of the
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509 computation involving assumed zeros is performed symbolically, not numerically.
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510
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511 This behavior not only facilitates the most straightforward and efficient
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512 implementation of algorithms, but also preserves certain useful invariants,
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513 like:
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514
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515 @itemize
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516 @item scalar * diagonal matrix is a diagonal matrix
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517
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518 @item sparse matrix / scalar preserves the sparsity pattern
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519
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520 @item permutation matrix * matrix is equivalent to permuting rows
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521 @end itemize
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522 all of these natural mathematical truths would be invalidated by treating
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523 assumed zeros as numerical ones.
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524
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525 Note that @sc{matlab} does not strictly follow this principle and converts
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526 assumed zeros to numerical zeros in certain cases, while not doing so in
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527 other cases. As of today, there are no intentions to mimic such behavior
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528 in Octave.
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529
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530 Examples of effects of assumed zeros vs. numerical zeros:
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diff changeset
531
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532 @example
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533 Inf * eye (3)
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534 @result{}
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535 Inf 0 0
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536 0 Inf 0
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537 0 0 Inf
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538
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539 Inf * speye (3)
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540 @result{}
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diff changeset
541 Compressed Column Sparse (rows = 3, cols = 3, nnz = 3 [33%])
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542
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543 (1, 1) -> Inf
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544 (2, 2) -> Inf
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545 (3, 3) -> Inf
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546
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547 Inf * full (eye (3))
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548 @result{}
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549 Inf NaN NaN
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550 NaN Inf NaN
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551 NaN NaN Inf
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552
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553 @end example
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554
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555 @example
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556 @group
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557 diag (1:3) * [NaN; 1; 1]
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558 @result{}
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559 NaN
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560 2
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561 3
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562
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563 sparse (1:3,1:3,1:3) * [NaN; 1; 1]
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564 @result{}
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565 NaN
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566 2
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567 3
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568 [1,0,0;0,2,0;0,0,3] * [NaN; 1; 1]
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569 @result{}
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570 NaN
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571 NaN
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572 NaN
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573 @end group
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574 @end example
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575