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<div class="ChapSects"><a href="chap6.html#X812CCAB278643A59">6 <span class="Heading">Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X802118FB7C94D6BA">6.1 <span class="Heading">Some operations for matrices</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X787B89237E1398B6">6.1-1 DirectSumDecompositionMatrices</a></span>
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<h3>6 <span class="Heading">Matrices</span></h3>
<p><a id="X802118FB7C94D6BA" name="X802118FB7C94D6BA"></a></p>
<h4>6.1 <span class="Heading">Some operations for matrices</span></h4>
<p><a id="X787B89237E1398B6" name="X787B89237E1398B6"></a></p>
<h5>6.1-1 DirectSumDecompositionMatrices</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumDecompositionMatrices</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>In June 2023 Hongyi Zhao asked in the Forum for a function to implement matrix decomposition into blocks. Such a function was then provided by Pedro García-Sánchez. Hongyi Zhao then requested that the function be added to <strong class="pkg">Utils</strong>. What is provided here is a revised version of the original solution, returning a list of decompositions.</p>
<p>This function is a partial inverse to the undocumented library operation <code class="code">DirectSumMat</code>. So if <span class="SimpleMath">L</span> is the list of diagonal decompositions of a matrix <span class="SimpleMath">M</span> then each entry in <span class="SimpleMath">L</span> is a list of matrices, and the direct sum of each of these lists is equal to the original <span class="SimpleMath">M</span>.</p>
<p>In the following examples, <span class="SimpleMath">M_6</span> is an obvious direct sum with <span class="SimpleMath">3</span> blocks. <span class="SimpleMath">M_4</span> is an example with three decompositions, while <span class="SimpleMath">M_8 = M_4 ⊕ M_4</span> has <span class="SimpleMath">16</span> decompositions (not listed).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M6 := [ [1,2,0,0,0,0], [3,4,0,0,0,0], [5,6,0,0,0,0], </span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0,0,9,0,0,0], [0,0,0,1,2,3], [0,0,0,4,5,6] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M6 );</span>
[ [ 1, 2, 0, 0, 0, 0 ],
[ 3, 4, 0, 0, 0, 0 ],
[ 5, 6, 0, 0, 0, 0 ],
[ 0, 0, 9, 0, 0, 0 ],
[ 0, 0, 0, 1, 2, 3 ],
[ 0, 0, 0, 4, 5, 6 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L6 := DirectSumDecompositionMatrices( M6 );</span>
[ [ [ [ 1, 2 ], [ 3, 4 ], [ 5, 6 ] ], [ [ 9 ] ], [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">M4 := [ [0,3,0,0], [0,0,0,0], [0,0,0,0], [0,0,4,0] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M4 );</span>
[ [ 0, 3, 0, 0 ],
[ 0, 0, 0, 0 ],
[ 0, 0, 0, 0 ],
[ 0, 0, 4, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L4 := DirectSumDecompositionMatrices( M4 );</span>
[ [ [ [ 0, 3 ] ], [ [ 0, 0 ], [ 0, 0 ], [ 4, 0 ] ] ],
[ [ [ 0, 3 ], [ 0, 0 ] ], [ [ 0, 0 ], [ 4, 0 ] ] ],
[ [ [ 0, 3 ], [ 0, 0 ], [ 0, 0 ] ], [ [ 4, 0 ] ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">for L in L4 do </span>
<span class="GAPprompt">></span> <span class="GAPinput"> A := DirectSumMat( L );; </span>
<span class="GAPprompt">></span> <span class="GAPinput"> if ( A = M4 ) then Print( "yes, A = M4\n" ); fi; </span>
<span class="GAPprompt">></span> <span class="GAPinput"> od;</span>
yes, A = M4
yes, A = M4
yes, A = M4
<span class="GAPprompt">gap></span> <span class="GAPinput">M8 := DirectSumMat( M4, M4 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( M8 );</span>
[ [ 0, 3, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 4, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 3, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 4, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L8 := DirectSumDecompositionMatrices( M8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( L8 ); </span>
16
</pre></div>
<p>The current method does not, however, catch all possible decompositions. In the following example the matrix <span class="SimpleMath">M_5</span> has its third row and third column extirely zero, and the only decomposition found has a <span class="SimpleMath">[0]</span> factor. There are clearly two <span class="SimpleMath">2</span>-factor decompositions with a <span class="SimpleMath">2</span>-by-<span class="SimpleMath">3</span> and a <span class="SimpleMath">3</span>-by-<span class="SimpleMath">2</span> factor, but these are not found at present.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M5 := [ [1,2,0,0,0], [3,4,0,0,0], [0,0,0,0,0],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [0,0,0,6,7], [0,0,0,8,9] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(M5);</span>
[ [ 1, 2, 0, 0, 0 ],
[ 3, 4, 0, 0, 0 ],
[ 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 6, 7 ],
[ 0, 0, 0, 8, 9 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">L5 := DirectSumDecompositionMatrices( M5 ); </span>
[ [ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0 ] ], [ [ 6, 7 ], [ 8, 9 ] ] ] ]
</pre></div>
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