Editing Hadamard matrix (section)

==Special cases==
Many special cases of Hadamard matrices have been investigated in the mathematical literature.

===Skew Hadamard matrices===
A Hadamard matrix ''H'' is ''skew'' if <math>H^\textsf{T} + H = 2I.</math>  A skew Hadamard matrix remains a skew Hadamard matrix after multiplication of any row and its corresponding column by −1.   This makes it possible, for example, to normalize a skew Hadamard matrix so that all elements in the first row equal 1.

Reid and Brown in 1972 showed that there exists a doubly regular [[tournament (graph theory)|tournament]] of order ''n'' if and only if there exists a skew Hadamard matrix of order ''n''&nbsp;+&nbsp;1.  In a mathematical tournament of order ''n'', each of ''n'' players plays one match against each of the other players, each match resulting in a win for one of the players and a loss for the other.  A tournament is regular if each player wins the same number of matches.  A regular tournament is doubly regular if the number of opponents beaten by both of two distinct players is the same for all pairs of distinct players.  Since each of the ''n''(''n'' − 1)/2 matches played results in a win for one of the players, each player wins (''n'' − 1)/2 matches (and loses the same number).  Since each of the (''n'' − 1)/2 players defeated by a given player also loses to (''n'' − 3)/2 other players, the number of player pairs (''i'', ''j'') such that ''j'' loses both to ''i'' and to the given player is (''n'' − 1)(''n'' − 3)/4.  The same result should be obtained if the pairs are counted differently: the given player and any of the ''n'' − 1 other players together defeat the same number of common opponents.  This common number of defeated opponents must therefore be (''n'' − 3)/4.  A skew Hadamard matrix is obtained by introducing an additional player who defeats all of the original players and then forming a matrix with rows and columns labeled by players according to the rule that row ''i'', column ''j'' contains 1 if ''i''&nbsp;=&nbsp;''j'' or ''i'' defeats ''j'' and −1 if ''j'' defeats ''i''.  This correspondence in reverse produces a doubly regular tournament from a skew Hadamard matrix, assuming the skew Hadamard matrix is normalized so that all elements of the first row equal 1.<ref>{{cite journal|last1=Reid|first1=K.B.|last2=Brown|first2=Ezra|title=Doubly regular tournaments are equivalent to skew hadamard matrices|journal=Journal of Combinatorial Theory, Series A |year=1972 |volume=12 |issue=3|pages=332–338 |doi=10.1016/0097-3165(72)90098-2|doi-access=free}}</ref>

===Regular Hadamard matrices===
[[Regular Hadamard matrices]] are real Hadamard matrices whose row and column sums are all equal. A necessary condition on the existence of a regular ''n''&thinsp;×&thinsp;''n'' Hadamard matrix is that ''n'' be a [[square number]]. A [[circulant]] matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be of square order. Moreover, if an ''n''&thinsp;×&thinsp;''n'' circulant Hadamard
matrix existed with ''n'' > 1 then ''n'' would necessarily have to be of the form 4''u''<sup>2</sup> with ''u'' odd.<ref>{{cite journal |first=R. J. |last=Turyn |title=Character sums and difference sets |journal=[[Pacific Journal of Mathematics]] |volume=15 |issue=1 |pages=319–346 |year=1965 |mr=0179098 |doi=10.2140/pjm.1965.15.319|doi-access=free }}</ref><ref>{{cite book |first=R. J. |last=Turyn |chapter=Sequences with small correlation |editor-first=H. B. |editor-last=Mann |title=Error Correcting Codes |publisher=Wiley |location=New York |year=1969 |pages=195–228 }}</ref>

===Circulant Hadamard matrices===
The circulant Hadamard matrix conjecture, however, asserts that, apart from the known 1&thinsp;×&thinsp;1 and 4&thinsp;×&thinsp;4 examples, no such matrices exist. This was verified for all but 26 values of ''u'' less than 10<sup>4</sup>.<ref>{{cite journal |first=B. |last=Schmidt |title=Cyclotomic integers and finite geometry |journal=[[Journal of the American Mathematical Society]] |volume=12 |issue=4 |pages=929–952 |year=1999 |doi=10.1090/S0894-0347-99-00298-2  |jstor=2646093 |doi-access=free |hdl=10356/92085 |hdl-access=free }}</ref>