MSc Mathematics Master Thesis
Hadamard matrices over ∗-rings
Maarten Havinga 22th June 2016
Supervisor: prof. dr. Lex Schrijver Second examiner: dr. G. Regts Daily supervisor: prof. dr. A. Brouwer
Korteweg-de Vries Insitute for Mathematics
Table of contents
Abstract 3
Introduction 3
1. Hadamard matrices 5
1.1 Hadamard matrices over ∗-rings 5
1.2 Hadamard matrices and ∗-homomorphisms 8
1.3 Constructions for H± 10 2. Conference matrices 16 2.1 Power matrices 17 2.2 Constructed families 19 Popular summary 23 References 24
Abstract
This thesis explores a generalized definition of Hadamard matrices over any ∗-ring and obtains constructions of families of Hadamard matrices over C for every Hadamard matrix over the quaternions H or a related ∗-ring. The main result (cf. Corollary 3) of this thesis is a construction of Hadamard matrices over a ∗-ring B for a Hadamard matrix H over a ∗-ring A and a unital ∗-homomorphism from A to a matrix ring over B. Moreover, this construction can be extended in many cases to obtain an affine family of Hadamard matrices over B (cf. Algorithm 4). The application for A = H and B the complex numbers is explored, yielding families of complex Hadamard matrices of size 2q + 2 with q + 1 parameters, where q is an arbitrary prime power.
Introduction
Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum information [14]. In general, they are useful objects in the theory of quantum information: they allow for constructing bases of unitary operators, bases of maximally entangled states and unitary depolarisers. The problem of constructing these three objects is equivalent in the sense that for every object of any of these three types one can construct an object of the other types, and also a teleportation scheme and a dense coding scheme[12]. In addition, they are useful for constructing quantum designs (a term including both SIC-POVMs and MU-bases) [13]. In fact, the task of finding (k +1) MU-bases is equivalent to finding a collection of k mutually unbiased Hadamards (MUHs). This is applicable in quantum tomography, for measuring the parameters characterizing a density matrix as precisely as possible. Mathematical applications of theory about Hadamard matrices include ∗-subalgebras of finite dimensional von Neumann algebras[8] and equiangular lines [5].
This thesis uses a general definition of Hadamard matrices over ∗-rings and strongly relates them through ∗-homomorphisms in Corollary 3. It also presents an algorithm, described in Corollary 3 up to Example 5, that constructs a family of 2n × 2n Hadamard matrices from certain n × n quaternion Hadamard matrices. This algorithm uses a ∗-homomorphism from the quaternions to 2 × 2 matrices over C, multiplied with √2 and a 2 × 2 unitary matrix over C. This algorithm is applied first to a family of 4 × 4 Hadamard matrices over the quaternions in Example 7. In Corollary 10 an easy series of quaternion Hadamard matrices is constructed from complex conference matrices, and the algorithm is applied to this series as well in Corollary 11. This yields families of complex
Hadamard matrices of size 2(q + 1) with q + 1 parameters for q any prime power, including Butson-type matrices of type lcm(d, 2) for d a divisor of q + 1.
Other results in this thesis: for H any square matrix over any ∗-ring an equation relating H∗H and G∗G for G the entrywise image of H under a ∗-homomorphism (cf. Theorem 2), a straightforward link between quaternion Hadamard matrices and com-plex Hadamard matrices of doubled size (Corollary 6), for A any ∗-ring and A[j]± an anticommutative ∗-ring extension a construction of Hadamard matrices over A[j]± from (a)symmetric conference matrices over A (cf. Theorem 8), a conjecture-based result on the possible equivalences of the constructed complex Hadamard matrices (cf. Conjecture 12) and finally a generalization of Di¸ta’s construction (cf. Lemma 1) to ∗-rings in Lemma 13. In short, this thesis offers a wholly new perspective on (complex) Hadamard matrices, using ∗-rings.
1. Hadamard matrices
1.1. Hadamard matrices over ∗-rings
Definition. A ∗-ring is a ring D endowed with a conjugation ∗ : D → D such that (ab)∗ = b∗a∗, (a + b)∗ = a∗+ b∗ and (a∗)∗ = a for all a, b ∈ D. A ∗-ring is called unital if there is an element 1 ∈ D with 1 · a = a · 1 = a for every a ∈ D. In this case, the group of units U (D) of D is the set {u ∈ D : u∗u = 1 = uu∗} which is a group under multiplication where every element u has an inverse u−1= u∗.
Example. The ring of complex numbers C with complex conjugation as ∗-operation.
Example. The ring of quaternions H+ = {a + bi + cj + dk : a, b, c, d ∈ R} with conjugation (a + bi + cj + dk)∗ = a − bi − cj − dk. In H+, multiplication is defined by distributiveness and by the equations ij = k = −ji, jk = i = −kj, ki = j = −ik and i2 = j2= k2= −1.
Example. The ring of split-quaternions H− = {a + bi + cj + dk : a, b, c, d ∈ R} with multiplication defined by distributiveness and by the equations ij = k = −ji, jk = −i = −kj, ki = j = −ik, k2 = j2 = 1 and i2 = −1, and endowed with ∗-operation
(a + bi + cj + dk)∗ = a − bi + cj + dk.
Example. A final example is the finite field Fq with q = p2k elements for p ∈ N prime
and k ∈ N, with ∗-operation 0∗ = 0 and x∗ = xpk for x 6= 0, since (xpk)pk = x(pk)2 = xp2k = xq = x and (a + b)p = p X n=0 p n ! anbp−n = ap+ bp (mod p)
so applied k times this gives (a + b)pk = apk+ bpk, the required additive property of the ∗-operation.
Most examples, however, can be found in rings of square matrices.
Notation. For A a matrix, we denote with A(i, j) the entry on the ith row and jth column. For A some expression in i and j we denote with (A)i,j the matrix B with B(i, j) = A.
Fact. Let D be a ∗-ring. For n ∈ N, the ring Mn(D) formed by the n × n matrices over
D endowed with the ∗-operation H∗ = (H(j, i)∗)i,j for H ∈ Mn(D) is also a ∗-ring. If D
is unital, then Mn(D) is also unital.
Definition. A Hadamard matrix over D is an n×n matrix H over D with H∗H = nIn
and H(i, j) ∈ U (D) (i.e. H(i, j)∗H(i, j) = 1) for all i, j. Here Indenotes the n × n identity
matrix over D.
P Pn(D) denotes the group of phased permutation matrices over D, the group of
matrices P with a unique nonzero entry for every column and row such that P∗P = I. This implies that the nonzero entries of P are in U (D).
Two Hadamard matrices H1, H2 are equivalent if there are phased permutation
ma-trices P1 and P2 such that P1H1P2 = H2. We write H1 ∼= H2.
We call a Hadamard matrix H dephased if all the entries of the first row and column are the unit 1 ∈ D. Every Hadamard matrix is equivalent to a dephased one, but not uniquely for n > 2.
We call a dephased Hadamard matrix H Butson-type of type d [1] if H(i, j)d = 1 for all i, j.
Write BH(d, n) for the collection of n × n Hadamard matrices that are Butson-type of type d. It is unknown in general for which n, d there exist matrices in BH(d, n). For instance, the Hadamard conjecture [6] that BH(2, n) 6= ∅ for n = 1, 2 or 0 (mod 4) is still open despite many efforts. Turyn shows in [10] how to construct matrices in BH(2, 4n) from a matrix in BH(4, 2n) 6= ∅. Clearly, BH(d, n) ⊆ BH(dm, n) for all m ∈ N.
Often, Hadamard matrices come in families:
Definition. [14] Suppose that R is a k-dimensional subspace of the real (n − 1) × (n − 1) matrices and H = exp( ˆH) a n × n Hadamard matrix over C, where exp(·) denotes the entrywise complex exponentiation of a matrix. Then if the set
H(R) = {exp( ˆH + i · 0 0 0 R
!
) : R ∈ R}
contains only Hadamard matrices that are mutually inequivalent, it is an affine family of Hadamard matrices with k parameters, stemming from H.
Lemma 1. Let M = (mij) be an d × d and N1, . . . , Nd be k × k dephased Hadamard
matrices over C with families of m, n1, . . . , nd parameters respectively. Then the block
matrix Q = m11N1 m12N2 . . . m1dNd m21N1 m22N2 . . . m2dNd .. . ... ... md1N1 md2N2 · · · mddNd
is a dephased Hadamard matrix with m +
d
X
i=1
ni+ (d − 1)(k − 1) parameters.
Proof. Cf. Di¸ta [3].
1.2. Hadamard matrices and *-homomorphisms
This article focuses on similar structures of Hadamard matrices over general ∗-rings and relates them through ∗-homomorphisms.
Definition. For A and B ∗-rings, a ∗-homomorphism ρ : A → B is a map from A to B such that ρ(a1+ a2) = ρ(a1) + ρ(a2), ρ(a1a2) = ρ(a1)ρ(a2) and ρ(a1)∗ = ρ(a∗1) for all
a1, a2 ∈ A. In other words, a ∗-homomorphism is a ring homomorphism that preserves
the ∗-operation. If A and B contain units eA and eB, such a ∗-homomorphism is called
unital if ρ(eA) = eB.
Theorem 2. Let A, B be ∗-rings and C a ∗-subring of Matk(A). Let H ∈ Matn(C)
and ρ : C → Matm(B) any ∗-homomorphism and take arbitrary Ki ∈ U (Matm(B)) for
i = 1, . . . , n, then for G1= (ρ(H(i, j))Kj)i,j and G2 = (Kiρ(H(i, j)))i,j we have
G1G∗1 = (ρ(HH∗(i, j)))i,j ∈ Matmn(B) and
G∗2G2 = (ρ(H∗H(i, j)))i,j ∈ Matmn(B)
Proof. G1G∗1= (ρ(H(i, j))Kj)i,j((ρ(H(i0, j0)Kj0)i0,j0)∗ which equals
(ρ(H(i, j))Kj)i,j(Kj∗0ρ(H(i0, j0))∗)j0,i0 = (ρ(H(i, j))Kj)i,j(Ki∗0ρ(H(i0, j0)∗))j0,i0
Using (XY )(i, j) =P
kXikYkj for any two matrices X and Y we obtain that this equals l X k=1 ρ(H(i, k))KkKk∗ρ(H(k, j0)∗) ! i,j0 = l X k=1 ρ(H(i, k)H∗(k, j)) ! i,j = (ρ l X k=1 H(i, k)H∗(k, j) !
)i,j = (ρ(HH∗(i, j)))i,j
Conversely, G∗2G2 = ((Kiρ(H(i, j)))i,j)∗(Kiρ(H(i, j)))i,j which equals
(ρ(H(i, j))∗Ki∗)j,i(Kiρ(H(i, j)))i,j = (ρ(H(i, j)∗)Ki∗)j,i(Kiρ(H(i, j)))i,j
= l X k=1 ρ(H(k, i)∗)Kk∗Kkρ(H(k, j)) ! = l X k=1 ρ(H∗(i, k))ρ(H(k, j)) ! = (ρ l X k=1 H∗(i, k)H(k, j) !
This theoretical result is easy once understood, but it gives myriads of connections between Hadamard matrices over different ∗-rings:
Corollary 3. Let A, B be unital ∗-rings and let C, D be unital ∗-subrings of Matk(A) and
Matm(B) respectively for k, m ∈ N. Let H ∈ Matn(C) with H∗H = nI for n invertible in
A, let ρ : C → D be a unital ∗-homomorphism and let Ki ∈ U (D) for i = 1, . . . , n. Then
G1 =
√
m(ρ(H(i, j))Kj)i,j and G2 =
√
m(Kiρ(H(i, j)))i,j are Hadamard over B if their
entries are in U (B).
Proof. ρ(nI) = nI due to ρ(I) = I, so we have just proven this formula for G2 in Theorem
2. The relation n−1H∗H = I proves that n−1H∗ = H−1, so n−1HH∗ = I and thus G1G∗1 = mnI by Theorem 2. Over Matmn(B) this implies that mn1 G∗1 = G
−1 1 so G
∗ 1G1 =
mnI.
Then if G1 and G2 have their entries in U (B), they are Hadamard matrices.
Algorithm 4. If H∗H = nI for H ∈ Matn(C) then the same holds for F = P1HP2
where P1, P2 ∈ P Pn(C). Applying Corollary 3 to F with D = Matmn(C) often gives affine
families of mn × mn Hadamard matrices over C. Proof. Applying Corollary 3 to F gives that G1 =
√
m(ρ(F (i, j))Kj)i,jand G2=
√
m(Kiρ(F (i, j)))i,j
are Hadamard matrices over C if their entries are in U (C). The fact that a part of these matrices can form an affine family of Hadamard matrices over C is proven by the case of Corollary 11.
In Corollary 3, for K ∈ U (D) fixed and any unit u in U (C) the choice Ki = (ρ(u))∗K
(for G1) gives ρ(u)Kj = K, and the choice Ki = K(ρ(u))∗ (for G2) gives Kiρ(u) = K.
Thus phasing H on the right side (P2) is equivalent to picking another Kj for column j
in the construction of G1, and phasing H on the left side (P1) is equivalent to choosing
another Ki for row i in the construction of G2. Also, permutations of H can be undone
in G1 and G2 by the inverse permutation on G1 or G2 as matrix over D. Therefore this
algorithm gives families of at most n + nd parameters (n for the phasing, nd for different K’s) , where d is the dimension of U (D) over R.
1.3. Constructions for H±
Example 5. Take C = Mat1(A) = A for either A = H+ or A = H−, D = Mat2(B),
B = C, Kj = q 1 2 1 1 1 −1 ! and %+(1) = 1 0 0 1 ! , %+(j) = 0 −1 1 0 ! %+(i) = 0 i i 0 ! , %+(k) = i 0 0 −i !
where the definition is completed by linearity over R, or
%−(1) = 1 0 0 1 ! , %−(j) = 0 1 1 0 ! %−(i) = i 0 0 −i ! , %−(k) = 0 i −i 0 !
with the same completion of the definition.
Then ρ+ : H+ → Mat2(C) and ρ− : H− → Mat2(C) are ∗-homomorphisms and Kj ∈ U (Mat2(C)). The entries of
√
2Kjρ+(x) and
√
2ρ−(x)Kj are units if x can be written as z1+z2
2 +
z1−z2
2 j for z1, z2 ∈ U (C). Likewise, the entries of
√
2ρ+(x)Kj and
√
2Kjρ−(x) are
units if x can be written as z1+ z2 2 + j z1− z2 2 , which equals z1+ z2 2 + z1− z2 2 j for z1, z2∈ U (C).
Proof. Since ρ± is linear over R, clearly ρ±(x1+ x2) = ρ±(x1) + ρ±(x2), and it is easy to
check that ρ±(x)∗= ρ±(x∗) for all x. Then due to the distributivity of multiplication, we only need to check ρ±(x1x2) = ρ±(x1)ρ±(x2) for x1, x2 ∈ {1, i, j, k} and since ρ(1) = I
we need only consider i, j and k. Clearly
ρ±(i)2 = ρ+(j)2 = ρ+(k)2 = −I = ρ(−1) and ρ−(j)2 = ρ−(k)2= I = ρ(1)
which corresponds to i2 = j2 = k2 = −1 in H+ and j2 = k2 = −i2 = 1 in H−. Also, ρ±(i)ρ±(j) = ρ±(k), and ρ±(k)ρ±(i) = ρ±(j). Moreover, ρ+(j)ρ+(k) = ρ+(i) while ρ−(j)ρ−(k) = −ρ−(i) = ρ−(−i) corresponding to jk = i in H+ and jk = −i in
H−. Thus ρ±(x1)ρ±(x2) = −ρ±(x2)ρ±(x1) for all x1, x2 ∈ {i, j, k}, so ρ+ and ρ− are
∗-homomorphisms.
Then to prove that the entries are units, writing x = a + bj for a, b ∈ C. √ 2Kjρ+(x) = a a a −a ! + b −b −b −b ! , √ 2ρ+(x)Kj = a a a −a ! + −b b b b ! , √ 2Kjρ−(x) = a a a −a ! + b b −b b ! and √ 2ρ−(x)Kj = a a a −a ! + b −b b b ! . Filling in a = z1+z2 2 and b = z1−z2 2 gives a + b = z1, a − b = z2, ±(a − b) = ±z2
and ±(a + b) = ±z1 which are all units and the entries of
√ 2Kjρ+(x) and √ 2ρ−(x)Kj. Likewise, filling in a = z1+z2 2 and b = z1−z2 2 gives a + b = z1, a − b = z2, (−a + b) = −z2
and (a + b) = z1 which are all units and the entries of
√
2ρ+(x)Kj and
√
2Kjρ−(x).
It is obvious that Kj∗Kj = KjKj∗= I since both equal Kj2 = I.
Corollary 6. If H1 and H2 are n × n Hadamard matrices over C, then
H1 H2
H2 −H1
! is 2n × 2n Hadamard over C if and only if 12(H1+ H2+ H1j − H2j) is Hadamard over H+.
Proof. Write C = 12(H1+ H2) and D = 12(H1− H2). If H = C + Dj is Hadamard over
H+, we can apply Corollary 3 with the mathematical objects as in Example 5 to obtain that G2 = (
1 1 1 −1
!
ρ+(H(i, j)))i,j is Hadamard over C. Now writing H(i, j) as c + dj,
1 1 1 −1 ! ρ+(c + dj) = c c c −c ! + d −d −d −d !
so for Pr the permutation of indices sending k to
k+1 2 + q+1 2 ((−1)k+ 1), we have PrG2Pr= C + D C − D C − D −(C + D) ! = H1 H2 H2 −H1 !
which is therefore Hadamard (Pr rearranges the entries into the list with the odd entries
first and the even entries after).
Conversely, suppose that H = H1 H2 H2 −H1
!
is Hadamard over C, then apparently HH∗ = 2nI so H1H2T − H2H1T = 0. Then
(C + Dj)(C + Dj)∗= CC∗+ DD∗− CjD∗+ DjC∗
which equals 12(H1H1∗+ H2H2∗) plus an expression times j. If we show that this expression
times j equals 0, we have shown that (C + Dj)(C + Dj)∗ = nI. Now 4(DjC∗ − CjD∗)
equals
H1jH1∗+ H1jH2∗− H2jH1∗− H2jH2∗− H1jH1∗+ H1jH2∗− H2jH1∗+ H2jH2∗
= 2(H1jH2∗− H2jH1∗) = 2(H1H2T − H2H1T)j
since jα = αj for all α ∈ C and H1H2T − H2H1T = 0 so this equals 0.
It remains to show that the entries of C + Dj are in U (H). However, they are of the form α = z1+z2 2 + z1−z2 2 j with z1, z2∈ U (C). Therefore 4αα∗ = (z1+ z2)(z1+ z2) + (z1− z2)(z1− z2) = 2|z1|2+ 2|z2|2+ z1z2+ z2z1− z1z2− z2z1 = 2|z1|2+ 2|z2|2
which equals 4 since z1, z2∈ U (C) so αα∗ = 1. The same holds for α∗α, so α ∈ U (H).
Example 7. In [2], Chterental and Dokovic give a parametrization of all 4 × 4 Hadamard matrices over H+ up to equivalence, including the two-parameter special family:
1 1 1 1 1 −1 b −b 1 a −x −z 1 −a −y −w : x = 12(1 + a + b − ab) y = 12(1 − a + b + ab) z = 12(1 + a − b + ab) w = 12(1 − a − b − ab)
where a ∈ U (C) and b = b1 + b2j with b21+ b22 = 1. It can be shown that the entries of
this family of matrices are of the form z1+z2
2 +
z1−z2
b = ±1, or if b = ±j. The case b = j gives a one-parameter family G1= 1 1 1 1 1 −1 j −j 1 a −x −z 1 −a −y −w : x = 12(1 + a + j − aj) y = 12(1 − a + j + aj) z = 12(1 + a − j + aj) w = 12(1 − a − j − aj)
where a ∈ U (C) can be chosen arbitrarily. All entries of G1can be written as z1+z2 2+z1−z2 2j:
for x, y, z, w with z1 = ±1 and z2 = ±a or z2 = ±1 and z1 = ±a. Thus Example 5 gives
that H1 =
√
2(Kiρ+(G1(i, j)))i,j for Ki =
1 1 1 −1
!
is an affine one-parameter family of 8 × 8 complex Hadamard matrices, namely
H1 = 1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 −1 −1 1 1 1 1 a a −1 −a −a −1 1 −1 a −a −a 1 −1 a 1 1 −a −a −1 a a −1 1 −1 −a a a 1 −1 −a
One extra parameter arises from multiplying G1 with phased permutation matrices
over H+ on the left side: for bi+ cij with ci, bi ∈ C and |ci|2+ |bi|2= 1 for all i we have
√ 2Kiρ+(bi+ cij) = bi bi bi −bi ! + ci −ci −ci −ci ! ,
and thus we need that |ci + bi|2 = 1 = |ci|2 + |bi|2 so cibi must be purely imaginary.
Multiplying (bi+ cij) with x, y, z or w gives that a = ±1 after some calculations so we
cannot phase the last two rows. Then we can write ci = ibiri for ri∈ R with the relation
|bi|2= (1 + r2
i)−1 and define qi = (1 + iri). This gives
√
2Kiρ+(bi+ cij) =
biqi biqi
biqi −biqi
and then it is clear from the formula above we can dephase the argument of bi and bi per
row. Then write pi = |bi|qi with |pi| = 1 and we obtain the family
H1 = p1 p1 p1 p1 p1 p1 p1 p1 p1 −p1 p1 −p1 p1 −p1 p1 −p1 p2 p2 −p2 −p2 p2 −p2 −p2 p2 p2 −p2 −p2 p2 −p2 p2 p2 −p2 1 1 a a −1 −a −a −1 1 −1 a −a −a 1 −1 a 1 1 −a −a −1 a a −1 1 −1 −a a a 1 −1 −a
which we can dephase further by multiplying with p1 and p1 per column and the 3rd and
4th row with p2 to obtain
H1 = 1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 1 −1 1 b −1 −b b −1 −b 1 1 −b −1 b −b 1 b −1 1 1 a a −1 −a −a −1 1 −1 a −a −a 1 −1 a 1 1 −a −a −1 a a −1 1 −1 −a a a 1 −1 −a
for b = p1p−22 which can be chosen as an arbitrary element of U (C).
However, this is all obtained from one choice of Kj for all j and one choice of
homo-morphism. Kj can be chosen as any unitary matrix:
Example. Take C and ρ± as in Example 5, but now Ki = I. Then the entries of
√
2Kiρ+(x) and
√
2ρ+(x)Ki are units if x ·
√
2 can be written as a1+ b2i + b1j + a2k where
a, b ∈ U (C) for a = a1+ a2i and b = b1 + b2i. Likewise, the entries of
√
2ρ−(x)Kj and
√
2Kiρ−(x) are units if x ·
√
2 can be written as a + bj for a, b ∈ U (C).
Proof. It is already proven that ρ± are ∗-homomorphisms. Writing x ·√2 = a1 + b1i +
b2j + a2k we obtain √ 2Kjρ+(x) = √ 2ρ+(x)Kj = √ 2ρ+(x) = a −b b a ! .
Likewise, writing√2x = a + bj we obtain√2ρ−(x) = a b b a
!
. Since a, b ∈ U (C) and a = a∗∈ U (C) 3 b∗ = b this suffices.
Thus, many families of Hadamard matrices can be obtained indeed by the construction of Theorem 2 using this single ∗-homomorphism.
2. Conference matrices
Many Hadamard matrices can be obtained from conference matrices:
Definition. An n × n matrix H over a ∗-ring A is called a conference matrix if the diagonal of H is 0 while all other entries are in U (A), and H∗H = (n − 1) · I. We call a conference matrix symmetric if HT = H (mind: not H∗ = H) and asymmetric if HT = −H.
Notation. As in Example 5, we will use ± to indicate two different constructions at once. The value of ± should be kept equal throughout the whole construction, and ∓ = −(±). Theorem 8. For A a ∗-ring, let A[j]± be the ring extension {a1+ a2j : a1, a2 ∈ A} over
A with j2 = ∓1 and ja = a∗j. Extend the ∗-operation to A[j]± by (aj)∗ = ∓ja∗ and its
additive property to make A[j] a ∗-ring. Then for H an n × n conference matrix over A and P ∈ P Pn(A) arbitrary, G = PTHP + jI is a Hadamard matrix over A[j]+ if H is
symmetric and over A[j]− if H is asymmetric.
Proof. Observe that αjα = jα∗α = j ∀α ∈ U (A) and thus G = PTHP + PTjIP = PT(H + jI)P . Then
G∗G = P∗(H∗H + H∗j + j∗H + j∗jI)P = P∗(nI + H∗j + j∗H)P .
Due to ja = a∗j for a ∈ A we get H∗j + j∗H = jHT+ j∗H. For HT = H we take A[j]+, so j∗ = −j and thus jHT + j∗H = 0. For HT = −H we take A[j]−, so j = j∗ and therefore we obtain jHT + j∗H = 0. Thus
G∗G = nP∗IP = nI.
The diagonal entries of G are j and j∗j = ∓j2 = ∓(∓1) = 1 so j ∈ U (A[j]±). The off-diagonal entries of G are units multiplied with other units and therefore units. Thus G is a Hadamard matrix over A[j]±.
2.1. Power matrices
There is an easy series of conference matrices over C which we approach as follows: Definition. Denote the entries of q × q matrices over Fq by the elements of Fq. The
power matrix of degree n is the q × q matrix Qn defined by Qn(a, b) = (b − a)n ∈ Fq.
Note that Q0 is the all-one matrix over Fq.
Definition. A homomorphism χ : F∗q → U (C) extended to Fq by χ(0) = 0 is a character
of Fq. For a matrix M over Fq, write χ(M ) for the matrix created by applying χ to M
entrywise.
Theorem 9. Let Fq be a finite field and Qn a power matrix where q − 1 - n. Then
χ(Qn)∗χ(Qn) = qI − 11T and χ(Qn)1 = 1T χ(Qn) = 0 over C, where 1 denotes the
all-ones vector.
Proof. For the off-diagonal elements of χ(Qn)∗χ(Qn) the theorem states, with 0 6= k ∈ Fq:
X
x6=0
χ((x + k)n)χ(x−n) = −1
This rewrites into X
x6=0
χ((1 + kx−1)n) = −1, and thus we are taking the sum X
y6=1
χ(yn) = X
y∈Fq
χ(y)n− 1. Since χ(0) = 0 always, for g ∈ F∗q a generator this sum is of the form q−2 X k=0 χ(g)kn− 1 = χ(g) (q−1)n− 1 χ(g)n− 1 − 1 where χ(g) n 6= 1 = χ(gq−1)n because χ is injective.
Therefore this equals −1.
For the diagonal elements of χ(Qn)∗χ(Qn) we have
X
x6=0
χ(xn)χ(x−n) =X
x6=0
χ(xnx−n) = (q − 1)χ(1)
which is q − 1. The part χ(Qn)1 = 1T χ(Qn) = 0 can be shown similarly: for g ∈ F∗q a
generator, X x=F∗ q χ(xn) = q−2 X k=0 χ(g)kn
Corollary 10. Let Fq be a finite field, χ an injective character and Qn a power matrix such that r = gcdq−1 (q−1,n) 6= 1. Then for |α| = 1, H ± α = 0 1T ±1 αχ(Qn) ! is a family of conference matrices over C, and the off-diagonal entries of H1+ are r-th roots of unity.
Proof. Observe that
(Hα∗Hα)±= q ±α1Tχ(Q n) ±α−1χ(Qn)∗1 χ(Qn)∗χ(Qn) + 11T ! = qI
since the case q − 1|n implies r = 1 which contradicts our assumption. The diagonal of χ(Qn) is zero and the diagonal of Hα± as well. The off-diagonal entries of H1+ are
r-th roots of unity since those entries equal χ(an) for some a ∈ F∗q, so their rth power is
χ(an)r= χ(anr) = χ(1) = 1 since q − 1 divides nr.
Then if either n or q is even, Hα+ is a symmetric complex conference matrix since QTn = Qn in that case. If both n and q are odd, Hα− is an asymmetric complex conference
matrix since then
QTn = (Qn(b, a))a,b= ((a − b)n)a,b= −((b − a)n)a,b= −Qn.
Remark. For q ≡ 1(mod 4) and n = q−12 , H1++iI and H−1+ +iI are Butson-type Hadamard matrices in BH(q + 1, 4). (see also Turyn [11])
Proof. For q ≡ 1 (mod 4), n = q−12 is even and the entries of H±1+ are real, therefore H±1+∗ = H±1+T = H±1+. Thus
(H±1+ + iI)∗(H±1+ + iI) = H±1+∗H±1+ + I + i(H±1+∗− H+
±1) = (q + 1)I
Theorem. (Paley [9]) Take n = q−12 for q an odd prime power and χ injective. If q = 3 (mod 4), I + H1− will be a real-valued Hadamard matrix. If q = 1 (mod 4), define the substitution ψ by ±1 → ± 1 1 1 −1 ! and 0 → 1 −1 −1 −1 !
and apply it on the entries of H1+ to obtain a 2(q + 1) × 2(q + 1) real-valued Hadamard matrix.
2.2. Constructed families
Let χ(Qn) be as in Corollary 10 and Hα±0 :=
0 1T 1 α0χ(Qn)
!
. Then, choosing any solution of the complex square root,
Proposition. Gα± = P Hα±0P + jI where P = diag1, α1−1/2, . . . , α−q1/2
for α0, . . . , αq ∈
U (C) is a Hadamard matrix over H±.
Proof. This is a special case of Theorem 8, since H±= C[j]± as can be easily verified. Corollary 11. The family of matrices F+(α) =√2(Kρ+(G+α(i, j)))i,j or F−(α) =
√
2(ρ−(G−α(i, j))K)i,j
is an affine family of complex Hadamard matrices with q + 1 parameters. Proof. Example 5 gives that for K =
q 1 2 1 1 1 −1 ! the matrix F+(α) or F−(α) is a complex Hadamard if the entries of G±α can be written as z1+z2
2 +
z1−z2
2 j. The diagonal
entries of G±α are all j which can be written as 1+(−1)2 +1−(−1)2 j. The off-diagonal entries are elements z of U (C) which can be written as z+z2 +
z−z
2 j. Thus F
±(α) forms a family
of complex Hadamard matrices with q + 1 parameters. These parameters are independent because for k > 0, αk only affects the (2k + 1)th and (2k + 2)th column and row of the
matrices F±(α), while the other parameters affect other columns and rows. Since α0
affects none of the entries of the 2 × 2 blocks on the diagonal, it is independent of the other parameters. Finally, since q > 1, the effects of α0, . . . , αq cannot be reversed, due to
their different effect on even versus odd columns (for F−(α)) or rows (for F+(α)). Write Pq,n = F±(1q+1) obtained by applying Corollary 11 to
0 1T
1 α0χ(Qn)
! where Qn is a matrix over Fq. This is well defined because (−1)qn determines the sign of ±.
Corollary. Pq,n is in BH(d, m(q + 1)) for d = lcm(r, 2) with r = gcdq−1(n,q−1).
Fact. Reviewing a paper from Dita [4] reveals an easier construction that produces equiv-alent matrices from the circulant complex conference matrix Hαo, by constructing
P HαoP + I P HαoP − I
P∗Hα∗oP∗− I −P∗Hα∗oP∗− I !
∼
= F±(α)
where I is the identity matrix of size q + 1. This matrix equals PrF±(α)Pr for Pr the
permutation of indices sending k to k+12 +q+12 ((−1)k+ 1), as in Corollary 6. However, taking Ki= q 1 2 1 −1 −1 −1 !
for several i instead of Ki =
q 1 2 1 −1 −1 −1 ! in Example 5 gives Hadamard matrices which are slightly different.
Example. Over F4 = F2[X]/(X2+ X + 1) we have χ(Q1) = 0 1 ω2 ω4 1 0 ω4 ω2 ω2 ω4 0 1 ω4 ω2 1 0 for χ
defined by χ(X) = ω2 with ω = eπi3 . Subsequently using Kj and ρ+ as in Example 5,
P4,1= exp( πi 3 0 0 0 0 0 0 0 0 0 0 0 3 3 0 3 0 3 0 3 0 0 3 0 3 0 0 2 2 4 4 0 0 3 3 0 3 4 1 2 5 0 3 0 0 0 3 4 4 2 2 0 0 0 3 3 3 2 5 4 1 0 3 2 2 4 4 0 3 0 0 0 0 4 1 2 5 3 3 0 3 0 3 4 4 2 2 0 0 0 3 0 0 2 5 4 1 0 3 3 3 )
in dephased form, where exp(·) denotes the entrywise exponent of a matrix. For αj ∈ U (C),
the family P4,1 takes the dephased form
1 1 1 1 1 1 1 1 1 1 1 −1 −α1 α1 −α2 α2 −α3 α3 −α4 α4 1 −1 α1 −α1 α0 α0 ω2α0 ω2α0 ω4α0 ω4α0 1 1 −1 −1 α2α−10 −α2α−10 ω4α3α0−1 ωα3α−10 ω2α4α−10 ω5α4α0−1 1 −1 α0 α0 α2 −α2 ω4α0 ω4α0 ω2α0 ω2α0 1 1 α1α−10 −α1α−10 −1 −1 ω2α3α0−1 ω5α3α−10 ω4α4α−10 ωα4α−10 1 −1 ω2α0 ω2α0 ω4α0 ω4α0 α3 −α3 α0 α0 1 1 ω4α1α−10 ωα1α−10 ω2α2α0−1 ω5α2α−10 −1 −1 α4α−10 −α4α−10 1 −1 ω4α0 ω4α0 ω2α0 ω2α0 α0 α0 α4 −α4 1 1 ω2α1α−10 ω5α1α−10 ω4α2α−10 ωα2α−10 α3α−10 −α3α−10 −1 −1
which is equivalent to the family D10(6) of the online catalogue [14] (missing the discrete parameter w, which can be obtained by adding the family P4,2).
With all families of matrices presented by Corollary 11, the question arises how many of them are (in)equivalent. Different injective characters can be identified by raising one
to the power l with gcd(l, q − 1) = 1 which gives the same result as applying χ to Qnl
instead of Qn. Thus it suffices to check the different power matrices only. For χ fixed,
Pq,n ∼= Pq,m requires lcm(2,gcdq−1(n,q−1)) = lcm(2,gcdq−1(m,q−1)) to make sure they are of the
same Butson type. But if either q−1 gcd(n,q−1) or
q−1
gcd(m,q−1) is odd, say n, with m being even,
then the primitive unitary roots of order 2gcdq−1
(n,q−1) in Pq,n appear on every lower left
entry of a 2 × 2 block, whereas in Pq,m they appear on the other 3 entries in half of the
blocks where they appear. This cannot be transformed into one another by multiplication with phased permutation matrices, thus gcd(n, q − 1) = gcd(m, q − 1).
We conjecture:
Conjecture. If P2∗Pq,nP1 = Pq,m for phased permutation matrices P1and P2 then R∗2(jI +
χ(Qn))R1= (jI + χ(Qm)) for phased permutation matrices R1, R2 over H±.
Unfortunately, we have not been able to prove this so far. It would yield:
Conjecture 12. Pq,m ∼= Pq,n if and only if m = prn (mod q − 1) for some r ∈ N and p
the unique prime divisor of q.
Proof. Assuming the first conjecture, we have that R∗2(jI + χ(Qn))R1 = (jI + χ(Qm)).
Since jI has less nonzero entries than χ(Qn) the subspaces C, jC ⊆ H±are invariant under
the phasing. Thus R∗2jR1 = jI which implies R2 = R1, which we will write as τ σ for τ a
phasing function and σ a permutation on Fq: (R∗1QnR1)xy = τ (x)−1τ (y)(Qn)σ(x)σ(y). Since
gcd(n, q − 1) = gcd(m, q − 1) we have lm = n (mod q − 1) for l ∈ N with gcd(l, q − 1) = 1 which implies
(Qn)x,x+y= yn= ylm = (Qm)xl,xl+yl∀a, b ∈ Fq
For y = 0 this gives σ(x) = xl mod =dfor all x ∈ Fq, where =d= {x : xd= 1} ⊆ Fq. Then
for x = 0 this gives τ (0)−1τ (y)χ(yn) = χ(ylm) for all y ∈ Fq. Since lm = n this gives that
τ is constant and we can choose R1= σ to be defined by σ(x) = xl.
Moreover, for all x, y ∈ Fq we have σ(x + y) = σ(x) + σ(y) mod =d so we can apply
Theorem 3 of Lenstra’s paper[7]: σx = a · αx + b for all x ∈ Fq, where a ∈ F∗q, b ∈ Fq
and α is a ring automorphism of Fq. In other words, α(x) := xs for s = pr a power of p
because those are the only ring automorphisms of Fq. Then since σ(xy) = σ(x)σ(y) we
have (axs+b)(ays+b) = axsys+b for all x, y ∈ Fq. Taking x = y = 1 gives (a+b)2= a+b,
thus a + b ∈ {0, 1}. Taking x = y = 0 gives b2 = b so b ∈ {0, 1}. Finally taking x = 1, y = 0 gives ab + b2 = b and subtracting b2 = b then gives ab = 0, and since a ∈ F∗q this
implies that b = 0. Therefore a = 1 so xl= σx = xs, thus l = s = pr (modulo q − 1) with m = prn as desired.
In the perspective of ∗-homomorphisms and ∗-rings, we conclude with an extension of Lemma 1:
Lemma 13. Let H be an n × n Hadamard matrix over a ∗-ring D, write N = {1, . . . , n} and let {θk}k∈N be Hadamard matrices of dimension b over D. Finally, let ρ : U (D) →
B ⊆ PPb(D) be a surjective ∗-homomorphism. Then the block matrix ϑ(H) with blocks
θkρ(H(j, k)) is a Hadamard matrix of dimension nb over D.
Proof. We have
(ϑ(H)∗ϑ(H)) = (bρ((H∗H)(j, k)))jk = (bρ(nδjk))jk = nbI
Since the entries of θkρ(u) are the product of elements of the Hadamard matrix θk and
nonzero entries of the phased permutation matrix ρ(u) for all k, they are in U (D). Thus the entries of ϑ(H) are in U (D) and thus it is a Hadamard matrix.
Now since a block of ϑ(H)∗ϑ(H) is a sum of θ∗kθkρ((H∗H)(j, k)), θkcan be multiplied on
the left side with phased permutation matrices for each k, effectively multiplying the rows of θk with elements of U (D). For k = 1 these multiplications are inverted by dephasing
ϑ(H), multiplying it on the left with a phased permutation matrix. Also, the first row of every θk comes on the first row of ϑ(H) and is therefore also dephased. That leaves b − 1
rows of n − 1 matrices that we can phase with effect in the construction, giving a family of Hadamard matrices stemming from ϑ(H) with (n − 1)(b − 1) more parameters than the summed amount of parameters of families stemming from {H}∪{θk(1) : k ∈ N } if the new
parameters are independent of the already existing parameters. But the new parameters are obtained by phasing θk, which is independent of the parameters of θk by definition.
Then they must also be independent in the larger matrix ϑ(H), and clearly they are independent of the parameters of θj, j 6= k. Finally, the parameters of H return in ϑ(H)
as multiplication parameters of blocks θk, of which the other parameters do not phase the
first row. Thus the parameters of H are also independent of the new parameters. They are also independent of the parameters of θk because those are independent of multiplication
Popular summary
Suppose that you want to make a triangle with the largest surface as possible with one fixed vertex on x = y = 0, the two other vertices to be chosen from the box {−1 ≤ x, y ≤ 1}. It is clear that there are 4 solutions that can be transformed into each other by rotations of either 90 or 180 degrees. A similar situation for 3 dimensions: to find tetraeders with the largest volume with one fixed vertex on x = y = z = 0 and the other three vertices to be chosen from the cube {−1 ≤ x, y, z ≤ 1}. Generalize this to general dimension n where all coordinates of all nonzero vertices have distance at most 1 to 0, and state that the (hyper)volume must equal the upper bound 1n√nn; then you have the problem of finding Hadamard matrices (the columns of the matrix are the nonzero vertices and the entries are the coordinates of the vertices).
A regular generalization is to use complex numbers a + bi for the coordinates of the vertices, where distance at most 1 to 0 translates into a2+ b2 ≤ 1. Here i is the square root of −1. In this thesis I explore a further generalization to general ∗-rings, of which the com-plex numbers form an example. For two ∗-rings A and B, the notion of a ∗-homomorphism from A to B exists: it is a map that identifies the structures of part of the ∗-ring A and part of the ∗-ring B. I prove that such a ∗-homomorphism can be used to construct many Hadamard matrices over B from a single Hadamard matrix over A, and provide two examples: one that constructs Hadamard matrices of dimension 8 with complex coordi-nates, and one that constructs Hadamard matrices of dimension 2(q + 1) with complex coordinates. Here q can be any prime number or any power of a prime number, with 2 as only exception.
For applications, suppose that you want to communicate something through a lot of noise. Then Hadamard matrices are useful because their columns - the nonzero vertices of the first paragraph - are perpendicular and maximally distantiated from each other (distance√2n). Therefore if the coordinates of the vertices (elements of the ∗-ring) are the basic symbols of the language, using vertices of a Hadamard matrix as the basic symbols of communication is very resistant to noise: if the noise changes one codeword (vertex) with less than half the distance to another vertex, the incurred error can be corrected.
References
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Algebra and its Applications 428, 1178-2001
[3] P. Di¸ta (2004), Some results on the parametrization of complex Hadamard matrices, J. Phys. A: Math. Gen. 37 5355
[4] P. Dita (2011), Circulant conference matrices for new complex Hadamard matrices, arXiv:1107.1338v1 [math-ph]
[5] C. Godsil and A. Roy (2009), Equiangular lines, mutually unbiased bases, and spin models, European J. Combin. 30, no. 1, 246-262
[6] J. Hadamard (1893), R´esolution d’une question relative aux d´eterminants. Bulletin des Sci-ences Math´ematiques 17: 240–246
[7] H.W. Lenstra Jr. (1990), Automorphisms of Finite Fields. Journal of Number Theory 34: 33-4
[8] R. Nicoara (2006), A finiteness result for commuting squares of matrix algebras, J. Operator Theory 55, 295–310
[9] R.E.A.C. Paley (1933). On orthogonal matrices, Journal of Mathematics and Physics 12: 311–320
[10] R. J. Turyn (1970), Complex Hadamard matrices, pp. 435-437 in Combinatorial Structures and their Applications, Gordon and Breach, London
[11] R. J. Turyn (1972), An infinite class of Williamson matrices, pp. 319-321 in Journal of Combinatorial Theory, Series A, Vol. 12 issue 3
[12] R F Werner (2001), All teleportation and dense coding schemes, J. Phys. A: Math. Gen. 34 7081
[13] G. Zauner (1999), Quantum Designs: Foundations of a non-commutative Design Theory, International Journal of Quantum Information (IJQI), Volume 9, Issue 1, February 2011, page 445-507
[14] Complex Hadamard matrices, a catalogue by W. Bruzda, W. Tadej and K. ˙Zyczkowski. http://chaos.if.uj.edu.pl/˜karol/hadamard/
