A trade in a complex Hadamard matrix is a set of entries which can be changed
to obtain a different complex Hadamard matrix. We show that in a real Hadamard
matrix of order $n$ all trades contain at least $n$ entries. We call a trade
rectangular if it consists of a submatrix that can be multiplied by some scalar
$c \neq 1$ to obtain another complex Hadamard matrix. We give a
characterisation of rectangular trades in complex Hadamard matrices of order
$n$ and show that they all contain at least $n$ entries. We conjecture that all
trades in complex Hadamard matrices contain at least $n$ entries.Comment: 9 pages, no figure

E.J. Billington

M. Newman

R. Craigen

T.S. Michael

W. Tadej

W.P. Orrick

English

arXiv

Padraig Ó Catháin

Ian M. Wanless

Crossref

Trades in Complex Hadamard Matrices

A trade in a complex Hadamard matrix is a set of entries which can be

changed to obtain a different complex Hadamard matrix. We show that in a real

Hadamard matrix of order n all trades contain at least n entries. We call a trade

rectangular if it consists of a submatrix that can be multiplied by some scalar

c 6= 1 to obtain another complex Hadamard matrix. We give a characterisation of

rectangular trades in complex Hadamard matrices of order n and show that they

all contain at least n entries. We conjecture that all trades in complex Hadamard

matrices contain at least n entrie

Ó Catháin, Padraig

Wanless, Ian M.

DCU Online Research Access Service

Trades in complex Hadamard matrices∗PADRAIG Ó CATHÁIN †IAN M. WANLESS ‡School of Mathematical SciencesMonash University, VIC 3800, AustraliaNo Institute GivenAbstract. A trade in a complex Hadamard matrix is a set of entries which can bechanged to obtain a different complex Hadamard matrix. We show that in a realHadamard matrix of order n all trades contain at least n entries. We call a traderectangular if it consists of a submatrix that can be multiplied by some scalarc 6= 1 to obtain another complex Hadamard matrix. We give a characterisation ofrectangular trades in complex Hadamard matrices of order n and show that theyall contain at least n entries. We conjecture that all trades in complex Hadamardmatrices contain at least n entries.2010 Mathematics Subject classification: 05B05, 42C15, 94A08A complex Hadamard matrix of order n is an n×n complex matrix with unimod-ular entries which satisfies the matrix equationHH† = nIn,where H† is the conjugate transpose of H. If the entries are real (hence ±1) thematrix is Hadamard. The notion of a trade is well known in the study of t-designsand Latin squares [1]. For a complex Hadamard matrix we define a trade to be aset of entries which can be altered to obtain a different complex Hadamard matrixof the same order. In other words, a set T of entries in a complex Hadamard matrixH is a trade if there exists another complex Hadamard matrix H ′ such that H andH ′ disagree on every entry in T but agree otherwise. If H is a real Hadamardmatrix, we would insist that H ′ is also real.Example 1. The 8 shaded entries in the Paley Hadamard matrix below form atrade. + + + + + + + ++ − − − + − + ++ + − − − + − ++ + + − − − + −+ − + + − − − ++ + − + + − − −+ − + − + + − −+ − − + − + + −∗This work was inspired by the discussion after Will Orrick’s talk at the ADTHM’14 work-shop, and most of the work was undertaken at the workshop. The authors are grateful tothe workshop organisers and to BIRS. Research supported by ARC grants FT110100065 andDP120103067.†E-mail: p.ocathain@gmail.com‡E-mail: ian.wanless@monash.eduIf each of the shaded entries is replaced by its negative, the result is anotherHadamard matrix.We use the word switch to describe the process of replacing a trade by a new setof entries (which must themselves form a trade). In keeping with the precedentfrom design theory, our trades are simply a set of entries that can be switched.Information about what they can be switched to does not form part of the trade(although it may be helpful in order to see that something is a trade). For realHadamard matrices there can only be one way to switch a given trade, since onlytwo symbols are allowed in the matrices and switching must change every entryin a trade. However, for complex Hadamard matrices there can be more than oneway to switch a given trade.Example 2. Let u be a nontrivial third root of unity. The following matrix is a7×7 complex Hadamard matrix. The shaded entries again form a trade; they canbe multiplied by an arbitrary complex number c of modulus 1 to obtain anothercomplex Hadamard matrix. This matrix is due originally to Petrescu [5], and isavailable in the online database [2].1 1 1 1 1 1 11 −u u −u2 −1 −1 −u1 u −u −1 −u2 −1 −u1 −u2 −1 u −u −u −11 −1 −u2 −u u −u −11 −1 −1 −u −u u −u21 −u −u −1 −1 −u2 uThe size of a trade is the number of entries in it. We say that a trade is rectangularif the entries in the trade form a submatrix that can be switched by multiplyingall entries in the trade by some complex number c 6= 1 of unit modulus. In acomplex Hadamard matrix each row and column is a rectangular trade. Thus thereare always 1× n and n× 1 rectangular trades. Similarly, in any real Hadamardmatrix we may exchange any pair of rows to obtain another Hadamard matrix.The rows that we exchange necessarily differ in exactly half the columns, so thisreveals a 2× n2 rectangular trade (and similarly there are alwaysn2 ×2 rectangulartrades in real Hadamard matrices). Less trivial trades were used by Orrick [4]to generate many non-equivalent Hadamard matrices of orders 32 and 36. Thesmaller of Orrick’s two types of trades was a 4× n4 rectangular trade that hecalled a “closed quadruple”. Closed quadruples are often but not always presentin Hadamard matrices. The trades just discussed all have size equal to the order nof the host matrix. The trade in Example 1 is a non-rectangular example with thesame property.Trades in real Hadamard matrices and related codes and designs have been stud-ied occasionally in the literature, either to produce invariants to aid with clas-sification or to produce many inequivalent Hadamard matrices. See [4] and thereferences cited there. In the complex case, trades are related to parameterisingcomplex Hadamard matrices, some computational and theoretical results are sur-veyed in [6].Throughout this note we will assume that H = [hi j] is a complex Hadamard matrixof order n. We will use ri to denote the i-th row of H. If B is a set of columns thenri,B denotes the row vector which is equal to ri on the coordinates B and zeroelsewhere. We use B for the complement of the set B.We now characterise rectangular trades. We will use〈· , ·〉for the standard Her-mitian inner product under which rows of a complex Hadamard matrix are or-thogonal.Lemma 1. Suppose that a trade T in a complex Hadamard matrix H can beswitched by multiplying the entries in T by a complex number c 6= 1 of unit mod-ulus.1. Let B be the set of columns in which row ri of H contains elements of T . If r jis a row of H that contains no elements of T then ri,B is orthogonal to r j,B.2. T forms a rectangular trade on rows A and columns B if and only if ri,B isorthogonal to r j,B for every ri ∈ A and r j /∈ A.Proof. First, since the rows of H are orthogonal, we have that0 =〈ri,r j〉=〈ri,B,r j,B〉+〈ri,B,r j,B〉Now, multiplying the entries in the set of columns B by c, we have that0 =〈cri,B,r j,B〉+〈ri,B,r j,B〉= c〈ri,B,r j,B〉+〈ri,B,r j,B〉Subtracting, we find that (c− 1)〈ri,B,r j,B〉= 0. Given that c 6= 1 the first claimof the Lemma follows.We have just shown the necessity of the condition in the second claim. To checksufficiency we just have to verify that any two rows ri,r j in A will be orthogonalafter multiplication by c. This follows from〈cri,B,cr j,B〉+〈ri,B,r j,B〉= |c|〈ri,B,r j,B〉+〈ri,B,r j,B〉=〈ri,B,r j,B〉+〈ri,B,r j,B〉= 0.It is of interest to consider the size of a smallest possible trade. For (real) Hadamardmatrices of order n we show that arbitrary trades have size at least n, with equal-ity obviously achievable in a variety of ways. Then we show that in the generalcase rectangular trades have size at least n. The question for arbitrary trades incomplex Hadamard matrices remains open.Theorem 1. Let H be a Hadamard matrix of order n. Any trade in H has size atleast n.Proof. Suppose that H ′ is a Hadamard matrix that differs from H in strictly fewerthan n entries. Without loss of generality, we assume that H is normalised, thatthe first row of H contains the smallest non-zero number, say k, of differencesbetween H and H ′, and that all differences between H and H ′ occur in the firstr rows. By construction there are at least rk entries in the trade, so rk < n. Nowconsider the submatrix S of H formed by the first k columns and the bottom n− rrows. By Lemma 1, we know that each row of S is orthogonal to the all onesvector. If follows that S contains (n− r)k/2 negative entries. But the first columnof S consists entirely of ones. So by the pigeon-hole principle, some other columnof S must contain at least(n− r)k2(k−1)>nk−n2(k−1)=n2negative entries. This column of H is not orthogonal to the first column, a contra-diction.Now we consider complex Hadamard matrices. The following lemma is the keystep in our proof.Lemma 2. Let H be a complex Hadamard matrix of order n, and B a set of kcolumns of H. If α is a non-zero linear combination of the elements of B then αhas at least d nk e non-zero entries.Proof. Without loss of generality, we can write H in the formH =(T UV W)where T contains the columns in B and the rows in which α is non-zero. Wewill identify a linear dependence among the rows of U , then use this and anexpression for the inner product of r1 and r2 to derive the required result. Weassume that there are t non-zero entries αi in α and that if t > 2 then they obey|α2|> |α1|> |αi| for 3 6 i 6 t. We need to show that t > d nk e.For any column c j not in B, we have that〈c j,α〉= 0 since the columns of Hare orthogonal. Thus every column of U is orthogonal to α , and so there existsa linear dependence among the rows of U , explicitly: h1 j = ∑ti=2−αiα−11 hi j, forany j /∈ B. In particular, this shows that indeed t > 2.Since H is Hadamard, we know that all of the hi j have absolute value 1, and thatrows of H are necessarily orthogonal:〈r1,r2〉=〈r1,B,r2,B〉+〈r1,B,r2,B〉=〈r1,B,r2,B〉+〈 t∑i=2−αiα−11 ri,B,r2,B〉=〈r1,B,r2,B〉+t∑i=2−αiα−11〈ri,B,r2,B〉Since〈r1,r2〉= 0 and〈ri,B,r2,B〉=−〈ri,B,r2,B〉, this means thatα2α−11〈r2,B,r2,B〉=〈r1,B,r2,B〉+t∑i=3αiα−11〈ri,B,r2,B〉(1)Now, each inner product〈ri,B,r2,B〉is a sum of k complex numbers of modulusone, and |αiα−11 |6 1 for i > 3. So the absolute value of the right hand side of (1)is at most (t− 1)k. In contrast, the absolute value of the left hand side of (1) is|α2α−11 |(n− k)> n− k. It follows that n− k 6 (t−1)k, and hence t > dnk e.In the special case that t = nk , we observe that n− k = t(k− 1). This forces|〈ri,B,r2,B〉| = k for i 6= 2, which implies that all of the ri,B are collinear, andhence that T is a rank one submatrix of H.Theorem 2. If H is a complex Hadamard matrix and T is a rectangular tradewith a rows and b columns then ab > n.Proof. Without loss of generality, T consists of the upper left a×b submatrix ofH. By hypothesis, γ1 = ∑16i6a ri and γc = ∑16i6a(cri,B + ri,B) are both orthog-onal to the space U spanned by the remaining n− a rows of H. Now considerγ1− γc, which is zero in any column not contained in T . Observe that the orthog-onal complement of U is a-dimensional, and that the initial a rows of H span thisspace: thus γ1− γc is in the span of these rows, Lemma 2 applies, and ab > n.Let H be a Fourier Hadamard matrix of order v, and suppose that t | v. Then thereexist t rows of H containing only tth roots of unity. Their sum vanishes on all butvt coordinates, so Lemma 2 is best possible.On the other hand, if H is Fourier of prime order p, the only vanishing sum ofpth roots is the complete one. So in this case, a linear combination of at most trows will contain at most t zero entries.Open questionsOn the basis of Theorem 1 and Theorem 2 we are inclined to think that the answerto the following question is negative:Question 1: Can there exist trades of size less than n in an n×n complex Hadamardmatrix?It would also be nice to know how “universal” the rectangular trades we havestudied are.Question 2: Is every trade in a complex Hadamard matrix a linear combinationof rectangular trades of the form characterised in Lemma 1?This work was motivated in part by problems in the construction of compressedsensing matrices [3]. Optimal complex Hadamard matrices for our constructionhave the property that linear combinations of t rows vanish in at most t compo-nents.Question 3: Describe other families of Hadamard matrices with the property thatno linear combination of t rows contains more than t zeros. Or, if such matricesare rare, describe families in which no linear combination of t rows contain morethan f (t) zeros for some slowly growing function f .References1. E. J. Billington, Combinatorial trades: a survey of recent results, in: W.D. Wal-lis (Ed.), Designs 2002: Further Computational and Constructive Design The-ory, Kluwer, Boston, 2003, pp. 47–67.2. W. Bruzda, W. Tadej and K. Życzkowski, Catalogue of complex Hadamardmatrices, http://chaos.if.uj.edu.pl/˜karol/hadamard/, re-trieved 10/09/2014.3. D. Bryant and P. Ó Catháin, An asymptotic existence result on compressedsensing matrices. submitted, March 2014.4. W. P. Orrick, Switching operations for Hadamard matrices, SIAM J. DiscreteMath. 22 (2008), 31–50.5. M. Petrescu, Existence of Continuous Families of Complex Hadamard Matri-ces of Certain Prime Dimensions and Related Results, PhD thesis, Universityof California, Los Angeles, (1997).6. W. Tadej and K. Życzkowski A Concise Guide to Complex Hadamard Matri-ces, Open Systems & Information Dynamics, 13 (2006), 133–177.

Trades in complex Hadamard matrices

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Trades in complex Hadamard matrices

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