<p>Binary <a href="https://en.wikipedia.org/wiki/Walsh_matrix" class="extiw" title="w:Walsh matrix">Walsh matrix</a> in sequency order (compare <a href="//commons.wikimedia.org/wiki/File:Natural_and_sequency_ordered_Walsh_16.svg" title="File:Natural and sequency ordered Walsh 16.svg"><b>this</b></a> file) as a <a href="https://en.wikipedia.org/wiki/Matrix_multiplication" class="extiw" title="w:Matrix multiplication">product</a>
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<ul><li>White (0) and red (1): result with modulo 2 operations</li></ul><ul><li>Gray numbers: result with normal operations</li></ul><p>The two matrices used as factors (which are <a href="https://en.wikipedia.org/wiki/Transpose" class="extiw" title="w:Transpose">transposed</a> to each other) correspond to the same <a href="https://en.wikiversity.org/wiki/Walsh_permutation" class="extiw" title="v:Walsh permutation">Walsh permutation</a> <i>wp</i>(6,9,1,10) = (0,6,9,15, 1,7,8,14, 10,12,3,5, 11,13,2,4).<br>
All 48 Walsh permutations whose compression vectors are permutations of (1,6,9,10) or (2,5,13,14) could have been used instead.<br><i>wp</i>(6,9,1,10) was chosen, because it is the only one with a <a href="https://en.wikipedia.org/wiki/Symmetric_matrix" class="extiw" title="w:Symmetric matrix">symmetric</a> compression matrix (shown in the top left corner).
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It is the square root of <a href="//commons.wikimedia.org/wiki/File:Gray_code_*_bit_reversal_16.svg" title="File:Gray code * bit reversal 16.svg"><i>wp</i>(8,12,6,3)</a>, which permutes the natural ordered into the sequency ordered Hadamard matrix.
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<span class="int-own-work" lang="en">Own work</span>