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=nI_{n}\$

Now: $HH^{\mathrm {T} }=nI_{n}\$

Matrix multiplication is not commutative.

When I tried to calculate $H^{\mathrm {T} }H=nI_{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 $HH^{\mathrm {T} }=nI_{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=nI_{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, $HH^{\mathrm {T} }=nI_{n}\$ does follow more directly from the definition as stated. The way it was previously follows from the fact that $H^{\mathrm {T} }=nH^{-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 $HH^{\mathrm {T} }=nI_{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.

 Fixed. The article now has a few more words, explicitly stating

"... where I_n is the n × n identity matrix and H^T 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 $2^{k\cdot 2^{k-1}}$, 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 Hadamar 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)