Editing Hadamard transform (section)

{{Short description|Involutive change of basis in linear algebra}}
{{redirect-distinguish|Walsh transform|Walsh matrix}}
{{Use American English|date=January 2019}}
[[File:1010 0110 Walsh spectrum (single row).svg|thumb|300px|The [[matrix multiplication|product]] of a [[Boolean function]] and a [[Hadamard matrix]] is its [[Walsh spectrum]]:<ref>Compare Figure 1 in {{cite conference
 | last1 = Townsend | first1 = W.J.
 | last2 = Thornton | first2 = M.A.
 | contribution = Walsh spectrum computations using Cayley graphs
 | doi = 10.1109/mwscas.2001.986127
 | publisher = IEEE
 | series = MWSCAS-01
 | title = Proceedings of the 44th IEEE 2001 Midwest Symposium on Circuits and Systems (MWSCAS 2001)| date = 2001
 | volume = 1
 | pages = 110–113
 | isbn = 0-7803-7150-X
 }}</ref><br>(1, 0, 1, 0, 0, 1, 1, 0) × H(8) = (4, 2, 0, −2, 0, 2, 0, 2)]]
[[File:1010 0110 Walsh spectrum (fast WHT).svg|thumb|300px|[[Fast Walsh–Hadamard transform]], a faster way to calculate the Walsh spectrum of (1, 0, 1, 0, 0, 1, 1, 0).]]
[[File:1010 0110 Walsh spectrum (polynomial).svg|thumb|300px|The original function can be expressed by means of its Walsh spectrum as an arithmetical polynomial.]]
The '''Hadamard transform''' (also known as the '''Walsh–Hadamard transform''', '''Hadamard–Rademacher–Walsh transform''', '''Walsh transform''', or '''Walsh–Fourier transform''') is an example of a generalized class of [[Fourier transform]]s. It performs an [[orthogonal matrix|orthogonal]], [[symmetric matrix|symmetric]], [[Involution (mathematics)|involutive]], [[linear operator|linear operation]] on {{math|2<sup>''m''</sup>}} [[real number]]s (or [[complex number|complex]], or [[hypercomplex number]]s, although the Hadamard matrices themselves are purely real).

The Hadamard transform can be regarded as being built out of size-2 [[discrete Fourier transform]]s (DFTs), and is in fact equivalent to a multidimensional DFT of size {{math|2 × 2 × ⋯ × 2 × 2}}.<ref name="kunz" />  It decomposes an arbitrary input vector into a superposition of [[Walsh function]]s.

The transform is named for the [[France|French]] [[mathematician]] [[Jacques Hadamard]] ({{IPA|fr|adamaʁ|lang}}), the German-American mathematician [[Hans Rademacher]], and the American mathematician [[Joseph L. Walsh]].