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| | Hi there! :) My name is Stevie, I'm a student studying Modern Languages and Classics from Naringal, Australia.<br><br>My homepage ... Fifa 15 Coin Generator ([http://Hongbo.me/?document_srl=43594 http://Hongbo.me/?Document_srl=43594]) |
| In [[mathematics]], a '''Hadamard matrix''', named after the French [[mathematician]] [[Jacques Hadamard]], is a [[square matrix]] whose entries are either +1 or −1 and whose rows are mutually [[orthogonal]]. In geometric terms, this means that every two different rows in a Hadamard matrix represent two [[perpendicular]] [[vector space|vector]]s, while in [[combinatorics|combinatorial]] terms, it means that every two different rows have matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional [[parallelepiped|parallelotope]] spanned by the rows of an ''n''×''n'' Hadamard matrix has the maximum possible ''n''-dimensional [[volume]] among parallelotopes spanned by vectors whose entries are bounded in [[absolute value]] by 1. Equivalently, a Hadamard matrix has maximal [[determinant]] among matrices with entries of absolute value less than or equal to 1 and so, is an extremal solution of [[Hadamard's maximal determinant problem]].
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| Certain Hadamard matrices can almost directly be used as an [[error-correcting code]] using a [[Hadamard code]] (generalized in [[Reed–Muller code]]s), and are also used in [[balanced repeated replication]] (BRR), used by [[statistician]]s to estimate the [[variance]] of a [[parameter]] [[estimator]].
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| ==Properties==
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| Let ''H'' be a Hadamard matrix of order ''n''. The transpose of ''H'' is closely related to its inverse. The correct formula is:
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| :<math> H H^{\mathrm{T}} = n I_n \ </math>
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| where ''I<sub>n</sub>'' is the ''n'' × ''n'' [[identity matrix]] and ''H''<sup>T</sup> is the [[transpose]] of ''H''. To see that this is true, notice that the rows of ''H'' are all orthogonal vectors over the field of real numbers and each have length <math>\sqrt n</math>. Dividing ''H'' through by this length gives an [[orthogonal matrix]] whose transpose is thus its inverse. Multiplying by the length again gives the equality above. As a result,
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| :<math> \operatorname{det}(H) = \pm n^{\frac{n}{2}}, </math> | |
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| where det(''H'') is the [[determinant]] of ''H''.
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| Suppose that ''M'' is a complex matrix of order ''n'', whose entries are bounded by |''M<sub>ij</sub>''| ≤1, for each ''i'', ''j'' between 1 and ''n''. Then [[Hadamard's inequality|Hadamard's determinant bound]] states that
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| :<math> |\operatorname{det}(M)| \leq n^{n/2}. </math>
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| Equality in this bound is attained for a real matrix ''M'' if and only if ''M'' is a Hadamard matrix.
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| The order of a Hadamard matrix must be 1, 2, or a multiple of 4.
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| ==Sylvester's construction==
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| Examples of Hadamard matrices were actually first constructed by [[James Joseph Sylvester]] in 1867. Let ''H'' be a Hadamard matrix of order ''n''. Then the partitioned matrix
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| :<math>\begin{bmatrix} H & H\\ H & -H\end{bmatrix}</math>
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| is a Hadamard matrix of order 2''n''. This observation can be applied repeatedly and leads to the following sequence of matrices, also called [[Walsh matrix|Walsh matrices]].
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| :<math>
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| H_1 = \begin{bmatrix}
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| 1 \end{bmatrix},
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| </math>
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| :<math>
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| H_2 = \begin{bmatrix}
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| 1 & 1 \\
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| 1 & -1 \end{bmatrix},
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| </math>
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| and
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| :<math>
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| H_{2^k} = \begin{bmatrix}
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| H_{2^{k-1}} & H_{2^{k-1}}\\
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| H_{2^{k-1}} & -H_{2^{k-1}}\end{bmatrix} = H_2\otimes H_{2^{k-1}},
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| </math>
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| for <math> 2 \le k \in N </math>, where <math> \left.\otimes\right. </math> denotes the [[Kronecker product]].
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| In this manner, Sylvester constructed Hadamard matrices of order 2<sup>''k''</sup> for every non-negative integer ''k''.<ref>J.J. Sylvester. ''Thoughts on inverse orthogonal matrices, simultaneous sign successions, and tessellated pavements in two or more colours, with applications to Newton's rule, ornamental tile-work, and the theory of numbers.'' [[Philosophical Magazine]], 34:461–475, 1867</ref>
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| Sylvester's matrices have a number of special properties. They are [[Symmetric matrix|symmetric]] and have [[Trace (linear algebra)|trace]] zero. The elements in the first column and the first row are all [[Positive number|positive]]. The elements in all the other rows and columns are evenly divided between [[sign (mathematics)|positive and negative]]. Sylvester matrices are closely connected with [[Walsh function]]s.
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| ==Alternative construction==
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| If we map the elements of the Hadamard matrix using the [[group homomorphism]] <math> \{1,-1,\times\}\mapsto \{0,1,\oplus\} </math>, we can describe an alternative construction of Sylvester's Hadamard matrix. First consider the matrix <math> F_n </math>, the <math> n\times 2^n </math> matrix whose columns consist of all ''n''-bit numbers arranged in ascending counting order. We may define <math> F_n </math> recursively by
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| :<math>
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| F_1=\begin{bmatrix}
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| 0 & 1\end{bmatrix}
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| </math>
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| :<math>
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| F_n=\begin{bmatrix}
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| 0_{1\times 2^{n-1}} & 1_{1\times 2^{n-1}} \\
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| F_{n-1} & F_{n-1} \end{bmatrix}.
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| </math>
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| It can be shown by induction that the image of the Hadamard matrix under the above homomorphism is given by
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| :<math>
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| H_{2^n}=F_n^{\rm T}F_n.
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| </math>
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| This construction demonstrates that the rows of the Hadamard matrix <math> H_{2^n} </math> can be viewed as a length <math> 2^n </math> linear [[Error Correcting Code|error-correcting code]] of [[Linear code#Popular notation|rank]] ''n'', and [[Linear code#Properties|minimum distance]] <math> 2^{n-1} </math> with [[Linear code#Popular notation|generating matrix]] <math> F_n. </math>
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| This code is also referred to as a [[Walsh code]]. The [[Hadamard code]], by contrast, is constructed from the Hadamard matrix <math> H_{2^n} </math> by a slightly different procedure.
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| ==Hadamard conjecture==
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| The most important open question in the theory of Hadamard matrices is that of existence. The '''Hadamard conjecture''' proposes that a Hadamard matrix of order 4''k'' exists for every positive integer ''k''.
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| A generalization of Sylvester's construction proves that if <math>H_n</math> and <math>H_m</math> are Hadamard matrices of orders ''n'' and ''m'' respectively, then <math>\scriptstyle H_n \otimes H_m</math> is a Hadamard matrix of order ''nm''. This result is used to produce Hadamard matrices of higher order once those of smaller orders are known.
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| Sylvester's 1867 construction yields Hadamard matrices of order 1, 2, 4, 8, 16, 32, etc. Hadamard matrices of orders 12 and 20 were subsequently constructed by Hadamard (in 1893).<ref>{{cite journal |first=J. |last=Hadamard |title=Résolution d'une question relative aux déterminants |journal=[[Bulletin des Sciences Mathématiques]] |volume=17 |issue= |pages=240–246 |year=1893 |doi= }}</ref> In 1933, [[Raymond Paley]] discovered a [[Paley construction|construction]] that produces a Hadamard matrix of order ''q''+1 when ''q'' is any [[prime number|prime]] power that is [[Congruence relation|congruent]] to 3 modulo 4 and that produces a Hadamard matrix of order 2(''q''+1) when ''q'' is a prime power that is congruent to 1 modulo 4.<ref>{{cite journal |first=R. E. A. C. |last=Paley |title=On orthogonal matrices |journal=[[Journal of Mathematics and Physics]] |volume=12 |issue= |pages=311–320 |year=1933 |doi= }}</ref> His method uses [[finite field]]s. The Hadamard conjecture should probably be attributed to Paley.
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| The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is 92. A Hadamard matrix of this order was found using a computer by [[Leonard Baumert|Baumert]], [[Solomon W. Golomb|Golomb]], and [[Marshall Hall (mathematician)|Hall]] in 1962 at [[JPL]].<ref>{{cite journal |first=L. |last=Baumert |first2=S. W. |last2=Golomb |first3=M., Jr. |last3=Hall |title=Discovery of an Hadamard Matrix of Order 92 |journal=[[Bulletin of the American Mathematical Society]] |volume=68 |issue=3 |pages=237–238 |year=1962 |doi=10.1090/S0002-9904-1962-10761-7 |mr=0148686 }}</ref> They used a construction, due to [[John Williamson (mathematician)|Williamson]],<ref>{{cite journal |first=J. |last=Williamson |title=Hadamard’s determinant theorem and the sum of four squares |journal=[[Duke Mathematical Journal]] |volume=11 |issue=1 |pages=65–81 |year=1944 |doi=10.1215/S0012-7094-44-01108-7 |mr=0009590 }}</ref> that has yielded many additional orders. Many other methods for constructing Hadamard matrices are now known.
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| In 2005, Hadi Kharaghani and Behruz Tayfeh-Rezaie published their construction of a Hadamard matrix of order 428.<ref>{{cite journal |first=H. |last=Kharaghani |first2=B. |last2=Tayfeh-Rezaie |title=A Hadamard matrix of order 428 |journal=Journal of Combinatorial Designs |volume=13 |year=2005 |issue=6 |pages=435–440 |doi=10.1002/jcd.20043 }}</ref> As a result, the smallest order for which no Hadamard matrix is presently known is 668. <!-- Anon contributor: please go to the article's talk page and discuss your objection to this claim; properly sourced material cannot be removed from Wikipedia without a good reason. -->
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| {{As of|2008}}, there are 13 multiples of 4 less than or equal to 2000 for which no Hadamard matrix of that order is known.<ref name="dokovic">{{Cite journal| doi=10.1007/s00493-008-2384-z| last=Đoković| first=Dragomir Ž| title=Hadamard matrices of order 764 exist| journal=Combinatorica| year=2008| volume=28| issue=4|pages=487–489}}</ref> They are:
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| 668, 716, 892, 1004, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, and 1964.
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| ==Equivalence of Hadamard matrices==
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| Two Hadamard matrices are considered [[equivalence relation|equivalent]] if one can be obtained from the other by negating rows or columns, or by interchanging rows or columns. Up to equivalence, there is a unique Hadamard matrix of orders 1, 2, 4, 8, and 12. There are 5 inequivalent matrices of order 16, 3 of order 20, 60 of order 24, and 487 of order 28. Millions of inequivalent matrices are known for orders 32, 36, and 40. Using a [[equivalence relation#Comparing equivalence relations|coarser]] notion of equivalence that also allows [[transpose|transposition]], there are 4 inequivalent matrices of order 16, 3 of order 20, 36 of order 24, and 294 of order 28.<ref>{{cite journal|last=Wanless|first=I.M.|title=Permanents of matrices of signed ones|journal=Linear and Multilinear Algebra |year=2005 |volume=53 |pages=427–433 |doi=10.1080/03081080500093990}}</ref>
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| ==Skew Hadamard matrices==
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| A Hadamard matrix ''H'' is ''skew'' if <math>H^{\rm T} + H= 2I. \, </math>
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| 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'' + 1.
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| ==Generalizations and special cases==
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| Many generalizations and special cases of Hadamard matrices have been investigated in the mathematical literature. One basic generalization is the [[weighing matrix]], a square matrix in which entries may also be zero and which satisfies <math>WW^{T}=wI</math> for some w, its weight. A weighing matrix with its weight equal to its order is a Hadamard matrix.
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| Another generalization defines a [[complex Hadamard matrix]] to be a matrix in which the entries are complex numbers of unit [[absolute value|modulus]] and which satisfies ''H H<sup>*</sup>= n I<sub>n</sub>'' where ''H<sup>*</sup>'' is the [[conjugate transpose]] of ''H''. Complex Hadamard matrices arise in the study of [[operator algebras]] and the theory of [[quantum computation]].
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| [[Butson-type Hadamard matrices]] are complex Hadamard matrices in which the entries are taken to be ''q''<sup>th</sup> [[roots of unity]]. The term "complex Hadamard matrix" has been used by some authors to refer specifically to the case ''q'' = 4.
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| [[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''×''n'' Hadamard matrix is that ''n'' be a perfect square. A [[circulant]] matrix is manifestly regular, and therefore a circulant Hadamard matrix would have to be of perfect square order. Moreover, if an ''n''×''n'' circulant Hadamard
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| 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 }}</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>
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| The circulant Hadamard matrix conjecture, however, asserts that, apart from the known 1×1 and 4×4 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 }}</ref>
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| ==Practical applications==
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| *[[Olivia MFSK]] – an amateur-radio digital protocol designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands.
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| *[[balanced repeated replication|Balanced Repeated Replication]] (BRR) – a technique used by statisticians to estimate the [[variance]] of a [[statistical estimator]].
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| *[[Coded aperture]] spectrometry – an instrument for measuring the spectrum of light. The mask element used in coded aperture spectrometers is often a variant of a Hadamard matrix.
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| *Feedback Delay Networks – Digital reverberation devices which use Hadamard matrices to blend sample values
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| *[[Plackett–Burman design]] of experiments for investigating the dependence of some measured quantity on a number of independent variables.
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| *[[Robust Parameter Design (RPD)|Robust parameter designs]] for investigating noise factor impacts on responses
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| ==See also==
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| *[[Hadamard transform]]
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| *[[Combinatorial design]]
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| *[[Quincunx matrix]]
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| ==Notes==
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| {{reflist|2}}
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| ==Further reading==
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| *{{cite book |first=S. |last=Georgiou |first2=C. |last2=Koukouvinos |first3=J. |last3=Seberry |chapter=Hadamard matrices, orthogonal designs and construction algorithms |pages=133–205 |title=Designs 2002: Further computational and constructive design theory |location=Boston |publisher=Kluwer |year=2003 |isbn=1-4020-7599-5 }}
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| *{{cite journal |first=K. B. |last=Reid |first2=E. |last2=Brown |title=Doubly regular tournaments are equivalent to skew Hadamard matrices |journal=J. Combin. Theory Ser. A |volume=12 |year=1972 |issue=3 |pages=332–338 |doi=10.1016/0097-3165(72)90098-2 }}
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| *{{cite journal |first=J. |last=Seberry |first2=B. |last2=Wysocki |first3=T. |last3=Wysocki |title=On some applications of Hadamard matrices |journal=Metrika |volume=62 |year=2005 |issue=2–3 |pages=221–239 |doi=10.1007/s00184-005-0415-y }}
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| *{{cite book |first=R. K. |last=Yarlagadda |first2=J. E. |last2=Hershey |title=Hadamard Matrix Analysis and Synthesis |year=1997 |location=Boston |publisher=Kluwer |isbn=0-7923-9826-2 }}
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| ==External links==
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| *[http://rangevoting.org/SkewHad.html Skew Hadamard matrices] of all orders up to 100, including every type with order up to 28;
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| * {{cite web | author=[[N. J. A. Sloane]]
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| |url=http://neilsloane.com/hadamard
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| |title=Library of Hadamard Matrices}}
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| *[http://www.iasri.res.in/webhadamard On-line utility] to obtain all orders up to 1000, except 668, 716, 876 & 892.
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| * {{cite journal|doi=10.1214/aoms/1177730883 |first1=Alexander M.
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| |last1=Mood
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| |journal=Annals of Mathematical Statistics
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| |year=1964
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| |volume=17 |number=4 | title=On Hotelling's Weighing Problem
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| |pages=432–446}}
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| *{{cite journal|first1=L. D. | last1=Baumert | first2=Marshall | last2=Hall
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| |title=Hadamard matrices of the Williamson type
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| |journal=Math. Comp. | year=1965 | volume=19 | number=91 | pages=442–447
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| |doi=10.1090/S0025-5718-1965-0179093-2 |mr=0179093}}
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| *{{cite journal | first1=J. M. |last1=Goethals | first2=J. J. | last2=Seidel
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| |title=A skew Hadamard matrix of order 36 | journal=J. Austral. math. Soc.
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| |year=1970 | volume=11 | number=3 | pages=343–344
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| |doi=10.1017/S144678870000673X }}
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| *{{cite journal| first1=Jennifer | last1=Seberry Wallis
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| |title=On the existence of Hadamard matrices
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| |year=1976 | journal=J. Combinat. Theory A | volume=21 | number=2 | doi=10.1016/0097-3165(76)90062-5
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| |pages=188–195}}
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| *{{cite journal|first1=Jennifer | last1=Seberry
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| |title=A construction for generalized hadamard matrices
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| |year=1980 | journal=J. Statist. Plann. Infer. | volume=4 | number=4 | doi=10.1016/0378-3758(80)90021-X
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| |pages=365–368}}
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| *{{cite journal|first1=Hiroshi | last1=Kimura
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| |title=New Hadamard matrix of order 24
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| |journal=Graphs and Combinatorics | year=1989 | volume=5 |
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| pages=235–242 | number=1|doi=10.1007/BF01788676}}
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| * {{cite journal| first1=Edward | last1=Spence
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| |title=Classification of hadamard matrices of order 24 and 28
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| |journal=Discr. Math.
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| |year=1995 | volume=140 | number=1-3 | pages=185–242 | doi=10.1016/0012-365X(93)E0169-5
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| }}
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| *{{ cite journal | first1=H. |last1=Kharaghani
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| |first2=B. | last2=Tayfeh-Rezaie
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| |title=A Hadamard matrix of order 428
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| |journal= J. Combin. Des.
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| |year=2005 | volume= 13 | pages=435–440 | number=6
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| |url=http://math.ipm.ac.ir/tayfeh-r/papersandpreprints/h428.pdf
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| |doi=10.1002/jcd.20043}}
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| * {{cite web | url = http://oeis.org/search?q=Hadamard+Matrix | title=Hadamard Matrix}} in [[OEIS]]
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| {{DEFAULTSORT:Hadamard Matrix}}
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| [[Category:Design theory]]
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| [[Category:Matrices]]
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| [[Category:Unsolved problems in mathematics]]
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