Talk:Hadamard matrix

Anon edits
''Hadi Kharaghani and Behruz Tayfeh-Rezaie announced on 21 June 2004 that they constructed a Hadamard matrix of order 428. As a result, the smallest order for which no Hadamard matrix is presently known is 668.''

Anon, would you please advise us what your objection is to this statement? Bearcat 03:49, 7 September 2006 (UTC)

You can actually download a text file of their matrix and verify it yourself, so what's the problem? Ntsimp 17:11, 12 September 2006 (UTC)

Hadamard matrices of order 764 exist

 * http://arxiv.org/abs/math.CO/0703312

Might be helpful. Melchoir 02:27, 13 March 2007 (UTC)

Balanced repeated replication
Maybe balanced repeated replication deserves its own page? Will Orrick 01:01, 21 May 2007 (UTC)

Mistake or not?
Previously: $$ H^{\mathrm{T}} H = n I_n \ $$

Now: $$ H H^{\mathrm{T}} = n I_n \ $$

Matrix multiplication is not commutative.

When I tried to calculate $$ H^{\mathrm{T}} H = n I_n \ $$, the result did not derive from the definition: I am to multiply items from columns of matrix H when rows (not columns) of matrix H are supposed to be orthogonal.

When one tries to calculate $$ H H^{\mathrm{T}} = n I_n \ $$, everything comes out logically. (IMHO: I could have been mistaken).

This page Mathworld page has it also in the same way.

But, if the definition would be given as follows: "In mathematics, a Hadamard matrix is a square matrix whose entries are either +1 or −1 and whose columns are mutually orthogonal.", then the following formula $$ H^{\mathrm{T}} H = n I_n \ $$ would be true.

Am I correct? :) —Preceding unsigned comment added by 82.131.55.64 (talk) 16:47, 8 July 2008 (UTC)

Actually both are true, but I agree that the way you have it now, $$ H H^{\mathrm{T}} = n I_n \ $$ does follow more directly from the definition as stated. The way it was previously follows from the fact that $$ H^{\mathrm{T}} = n H^{-1} \ $$ and therefore commutes with $$H$$. This, of course, is not quite as immmediate. Will Orrick (talk) 19:34, 8 July 2008 (UTC)

Plain English
One of the things I hate about mathematical notation (especially in an Encyclopedia!) is that the definitions of the notations used are not immediately obvious to someone who's forgotten their math (or never learned it) and doesn't have an easily readable notation cheat sheet on hand.

From context, I'm assuming $$ H H^{\mathrm{T}} = n I_n \ $$ (thanks to the person above for the math markup) means "The matrix H noncommutatively multiplied with the matrix H to the Tth power (where the variable[?] T isn't defined) is equal to n times the nxn identity matrix. Is this correct?  What does $$ \operatorname{det} $$ mean?

Could those who write these articles begin appending plain (mathematical) language descriptions for those who haven't used matrix algebra (or whatever) in the previous decade (or ever)? I'd greatly appreciate it, I'm sure others would as well. --99.14.107.65 (talk) 05:04, 27 October 2008 (UTC)


 * I agree that we should wp:Make technical articles accessible by adding a few sentences to wp:explain jargon and make things more wp:obvious to people who don't already know everything about the topic, attempting to use plain English.
 * . The article now has a few more words, explicitly stating
 * "... where In is the n &times; n identity matrix and HT is the transpose of H. Consequently the determinant of H equals ...",
 * and linking to articles that go into more depth on those notations. Does that answer these specific questions? --68.0.124.33 (talk) 05:36, 20 February 2009 (UTC)

Determinant of Sylvester matrix
The determinant should be $$2^{k\cdot2^{k-1}}$$, but as this is just the usual dependence of the determinant of a Hadamard matrix on its size, it probably doesn't merit special mention. Will Orrick (talk) 17:31, 16 January 2009 (UTC)
 * Yup, that's my bad. Didn't see the bit on determinants above.  Have a good day, Robinh (talk) 20:39, 16 January 2009 (UTC)

Williamson type Hadamard matrices
It is a common mistake that Williamson type construction is from Williamson. It is actually from Golomb and two other co-authors of the paper in 1962. The construction is named after Williamson because Williamson later proved that such construction can be generalized for many other (possibly, infinitely many) values of n.

Hysong88 (talk) 01:50, 9 September 2009 (UTC)


 * This is not true. Williamson's construction dates from 1944.  Baumert, Golomb, and Hall cite Williamson's work in their paper.  I will add literature references to the article. Will Orrick (talk) 19:50, 10 September 2009 (UTC)

Circulant Hadamard matrices
The word "order" in the discussion of circulant Hadamard matrices did refer to the size of the matrices. The definition of "circulant" being used here is that each row of the matrix is rotated one unit to the right relative to the row above it. I have updated the article, pointing the the Wikipedia article where "circulant" is defined, and expanding a bit on what is known about the conjecture. —Preceding unsigned comment added by Will Orrick (talk • contribs) 23:55, 24 October 2009 (UTC)

images
Hello. Can anybody add a example image of some rather small Hadamard matrix? Smth. like mathworld.wolfram.com/HadamardMatrix.html or even better one http://www.math.pitt.edu/~egw1/Hadamard8.jpg not with '+' and '-', but with black and white squares. `a5b (talk) 23:03, 9 January 2010 (UTC)

Equivalence and transposition
I have reverted the edit that added transposition to the list of operations under which Hadamard matrices are considered equivalent. Obviously there are different possible definitions of equivalence, but if transposition is added to the list, then the numbers of equivalence classes listed in the article become wrong. For example, for 16×16 Hadamard matrices there are five equivalence classes up to row/column negation/permutation, but only four equivalence classes up to row/column negation/permutation and transposition. Will Orrick (talk) 16:17, 5 February 2010 (UTC)

For which natural numbers is the Hadamard conjecture known?
In the article it says:

''"As a result, the smallest order for which no Hadamard matrix is presently known is 668." ''

and it says:

''"As of 2008[update], there are 13 integers n less than or equal to 500 for which no Hadamard matrix of order 4n is known.[6] They are: 167, 179, 223, 251, 283, 311, 347, 359, 419, 443, 479, 487, 491." ''

Isnt that a contradiction? --131.234.106.197 (talk) 14:38, 18 January 2011 (UTC)


 * The two statements agree. The size of a Hadamard matrix must be a multiple of 4 — sizes 1 and 2 being the sole exceptions.  "Order" in the first statement refers to the actual matrix size.  The n in the second statement is a quarter of the matrix size.  One quarter of 668 is 167, which is the first number on the list of orders for which no Hadamard matrix is known.  Perhaps it's worth trying to clarify the article slightly to avoid confusion. Will Orrick (talk) 20:33, 18 January 2011 (UTC)

Circulant Hadamard conjecture
I've reverted the edit of 6 October 2011 by user 195.176.20.45. It is not clear that the "post-publication revision" has been refereed or accepted for publication anywhere, which, in my view, means that the claimed result is not yet suitable for inclusion in Wikipedia. The edit contained the somewhat odd sentence, "The version that appeared mistakenly contained an older version of the manuscript whose Section 3.3 had a flaw." By my reading, this is suggesting that it is not the original published article that was in error, but rather the form in which it appeared that was a mistake—as though the publisher accidentally printed a rough draft rather than the finished paper. If true, this would be an extraordinary story, one that should be backed up, at the very least, by a statement from the publisher. I have checked the website of the Bulletin of the London Mathematical Society; it does not appear to contain a correction or retraction of the article. On the other hand, the phrase "post-publication revision", which also appears in the comment line of the arxiv posting, suggests a revision made after publication, implying a different sequence of events. If more information comes to light, I would be in favor of restoring the reverted edits. Will Orrick (talk) 19:12, 6 October 2011 (UTC)

Enough with the proofs of the Hadamard circulant conjecture! While I have great respect for people who are attacking such difficult problems, I would point out that Wikipedia is a resource for non-experts seeking information about what is known about a topic, not a place for advertising recent preprints whose validity can only be confirmed by experts. Once the results in these papers become widely known to workers in the field, as they surely will, assuming they check out, it would be entirely appropriate to summarize them in the Wikipedia article. Will Orrick (talk) 22:50, 11 October 2011 (UTC)

Attribution of Hadamard Conjecture
An editor recently put a tag on the statement that Paley should probably be given the attribution for the Hadamard conjecture. I tried to locate a citation. Several texts cite Hadamard's 1893 paper for this, but I'm not actually convinced. Part of my hesitation is that in his 1933 paper, Paley proved that the order was 1,2,or 4n. If Hadamard didn't know this result, what would he have conjectured in 1893? Can someone look at Hadamard's paper to see what is actually there? Bill Cherowitzo (talk) 03:33, 7 April 2014 (UTC)


 * Hadamard's paper is available online at http://gallica.bnf.fr/ark:/12148/cb34348304d/date. You'll need to look at the 1893 volume, page 397.  This is the paper where Hadamard proves his determinant bound.  Here is my summary of what's in the paper:


 * After proving the bound, Hadamard notes that it is attained by Vandermonde matrices whose elements are the roots of unity. For size 3, he notes, Sylvester proved that, up to obvious transformations, this is the only maximum-determinant matrix. For size 4, Hadamard exhibits a one-parameter family of non-equivalent complex Hadamard matrices.


 * He then turns to the real case, noting that Sylvester's Kronecker-product construction gives maximum determinants for power-of-2 sizes. In answer to your question, he then indeed does prove that the size must be a multiple of 4 (if greater than 2).  He next exhibits the constructions he found for sizes 12 and 20.


 * At this point he makes the remark, "Il y a donc lieu de se demander quelles sont les valeurs de n pour lesquelles existent des déterminants maximum à éléments réels." (My translation: It is therefore appropriate to ask what are the values of n for which there exist maximum determinants with real elements.)  There is a footnote here, which reads, "Les déterminants maximum que nous venous de former pour n=12 et n=20 mettent encore un fois en évidence l'arbitraire que comporte la question actuelle; car il est clair que ces nouveaux détermiants maximum ne peuvent se déduire des procédés donnés au n° 6."  (My translation: The maximum determinants that we have just constructed for n=12 and n=20 put once more in evidence the arbitrariness that the present question involves; for it is clear that these new maximum determinants cannot be deduced from the procedures given in Section 6.  ("These procedures" are essentially Sylvester's Kronecker product construction as modified by Hadamard to allow multiplication by a unit of a fixed row in all of the submatrices contained in a given column of the array.))


 * After this remark, Hadamard concludes the paper with the following sentence, which poses what is now known as Hadamard's maximum determinant problem, "De plus, on peut rechercher, pour les autres valeurs, quel est le plus grand module que puisse atteindre le déterminant lorsque'on impose aux éléments la condition d'être réels." (My translation: In addition, one can study, for other values, what is the largest modulus that can be attained by the determinant when one imposes on its elements the condition of being real.)


 * There is no explicit (or, in my opinion, implicit) statement of the Hadamard conjecture in any of this. This doesn't surprise me: Hadamard only had powers of 2 and sizes 12 and 20 (and products of powers of these sizes).  There wasn't much evidence one way or the other at that point.  I've actually never seen the conjecture referred to as "Hadamard's conjecture" (as the anonymous editor does).  It's either the "Hadamard matrix conjecture" or the "Hadamard conjecture", which I've always interpreted as "the conjecture about Hadamard matrices" rather than as "the conjecture made by Hadamard".  Are there actually books that attribute the conjecture to Hadamard? Will Orrick (talk) 20:31, 7 April 2014 (UTC)


 * I also wanted to mention that the Paley construction article contains a quotation of the line from Paley's article where he conjectures that Hadamard matrices exist of every allowed size. Will Orrick (talk) 01:14, 8 April 2014 (UTC)

Thanks Will, that is pretty much what I suspected was true. Given what he knew about, it would have been a very bold conjecture on Hadamard's part. I have also never seen the possessive form of the title of the conjecture. What threw me off were two references which attributed the conjecture to Hadamard (Handbook of Combinatorial Designs, p. 274 and Doug Stinson's Combinatorial Designs –from which I quote–"It is a famous open conjecture, first stated by Jacques Hadamard in 1893, that there exists a Hadamard matrix of every order n ≡ 0 (mod 4)." p.74) As to my statement about Paley's contribution, that came from Dembowski's Finite Geometry, where he attributes the result to Paley and also to Ryser, but doesn't mention Hadamard. I wonder if this is just another instance of that French/German rivalry where they selectively forget to mention the contributions coming from the other country. Thanks again. Bill Cherowitzo (talk) 02:20, 8 April 2014 (UTC)