Threeparameter complex Hadamard matrices of order 6

Introduction

Complex Hadamard matrices have turned out hard to classify, with current classifications being incomplete for order 6 and higher. For order 6, there is evidence for a four-parameter family , but up till now only zero-, one- and two-parameter subfamilies have been obtained on closed form, as reviewed in . Recent progress includes the construction of three two-parameter, nonaffine families that contain the one-parameter families as subfamilies, and has resulted in an overall picture of five, partially overlapping, two-parameter families of complex Hadamard matrices of this order.

A further step towards a more comprehensive classification was taken in , where it was shown that any complex Hadamard matrix of order 6 is equivalent to (or equals) a Hadamard matrix for which either all (the $H_{2}$-reducible case) or none of its nine $2\times 2$ submatrices are Hadamard. In the present paper, a complete characterization of the $H_{2}$-reducible Hadamard matrices is given. The result is a three-parameter family which has all the previously known (one- and) two-parameter families as subfamilies.

Preliminaries

An $N\times N$ matrix $H$ with complex elements $h_{ij}$ is Hadamard if all elements have modulus one, $|h_{ij}|=1$, and if

(the unitarity constraint), where $E$ is the unit matrix in $N$ dimensions. Two Hadamard matrices are termed equivalent, $H_{1}\sim H_{2}$, if they can be related through

where $D_{1}$ and $D_{2}$ are diagonal unitary matrices, and $P_{1}$ and $P_{2}$ are permutation matrices. A set of equivalent Hadamard matrices can be represented by a dephased matrix, with ones in the first row and the first column.

The present paper will be concerned with Hadamard matrices which are reducible in the following sense.

A complex Hadamard matrix of order 6 is $H_{2}$-reducible if it is equivalent to a Hadamard matrix for which all the nine $2\times 2$ submatrices are Hadamard.

$H_{2}$-reducible Hadamard matrices are more prevalent than might be thought. The quite general nature of these matrices is illustrated by the following theorem that was proven in .

Let $H$ be a complex Hadamard matrix of order 6, with elements $h_{ij}$, $i,j=$1,…,6. If there exists an order 2 submatrix \left(\begin{array}[]{cc}h_{ij}&h_{ik}\\ h_{lj}&h_{lk}\end{array}\right) that is Hadamard, then $H$ is $H_{2}$-reducible.

Since the submatrix referred to in Theorem 2 has the (unique) dephased form

$H_{2}$-reducible Hadamard matrices are easily identified:

Let H be a complex Hadamard matrix of order 6. H is $H_{2}$-reducible if, and only if, its dephased form has at least one element equal to -1.

It follows from the corollary that all the currently known one- and two-parameter families in six dimensions ($F_{6}^{(2)},\,(F_{6}^{(2)})^{T}$,$D_{6}^{(1)}$ , $B_{6}^{(1)}$ , $M_{6}^{(1)}$ , $X_{6}^{(2)},\,(X_{6}^{(2)})^{T}$ and $K_{6}^{(2)}$ , in the notation of ) are families of $H_{2}$-reducible Hadamard matrices. On the other hand, the single, isolated matrix $S_{6}^{(0)}$ is not $H_{2}$-reducible.

A general, $H_{2}$-reducible Hadamard is equivalent to a Hadamard matrix on the dephased form (see )

where each of the (Hadamard) matrices $Z_{i}$ is fully determined by a single complex number $z_{i}$ of modulus one, $|z_{i}|=1$,

and where $a,\,b,\,c$ and $d$ are Hadamard matrices of order 2. Not all matrices of the general form (2.4) will be Hadamard, and the additional conditions on the matrix elements will now be investigated.

The unitarity constraints

In order to develop an exhaustive parametrization of the $H_{2}$-reducible Hadamard matrices on the standard form (2.4), the unitarity constraints on $H$ and its submatrices are first explored. In a second step, the additional constraints imposed by the unimodularity of the elements of $H$ are investigated.

Let $e$ be the unit matrix in two dimensions.

Let $H$ be an $H_{2}$-reducible Hadamard matrix on the standard form (2.4). Then

and where the $2\times 2$ matrix $\Lambda$ is unitary, $\Lambda^{\dagger}\Lambda=\Lambda\Lambda^{\dagger}=e$, and self-adjoint, $\Lambda^{\dagger}=\Lambda$.

In (2.4), let $a=\frac{1}{2}Z_{3}AZ_{1}$, $b=\frac{1}{2}Z_{3}BZ_{2}$, $c=\frac{1}{2}Z_{4}CZ_{1}$ and $d=\frac{1}{2}Z_{4}DZ_{2}$. In terms of $A$, $B$, $C$ and $D$, the full set of unitarity constraints on $H$ take the form

Note that these conditions are independent of $z_{1},\,z_{2},\,z_{3}$ and $z_{4}$. It follows from (3.1) that $D=A$ and $C=B$. The relations (3.2) can therefore be reduced to

In view of the constraint $A+B=-F_{2}$ (from (3.1)), the first two of these relations are always satisfied. In terms of $\Lambda\equiv-iF_{2}(A-B)/(2\sqrt{3})$, the last two relations imply that $\Lambda$ is unitary, $\Lambda^{\dagger}\Lambda=\Lambda\Lambda^{\dagger}=e$. Finally, by assumption, $a^{\dagger}a=b^{\dagger}b=2e$ so that $A^{\dagger}A=B^{\dagger}B=2e$. As a result, $(A+B)^{\dagger}(A-B)+(A-B)^{\dagger}(A+B)=0$ or, in terms of $\Lambda$, $\Lambda-\Lambda^{\dagger}=0$. Solving for $A$ and $B$ in terms of $F_{2}$ and $\Lambda$ completes the proof.∎

If a $2\times 2$ matrix $\Lambda$ is unitary and self-adjoint, either $\Lambda=\pm e$ or

with $\theta\in[0,2\pi)$ and $\phi\in[0,2\pi)$.

Since $\Lambda$ is self-adjoint, its diagonal elements $\Lambda_{11}$ and $\Lambda_{22}$ are real, and $\Lambda_{21}=\bar{\Lambda}_{12}$. Furthermore, since $\Lambda$ is unitary,

The off-diagonal elements vanish if either $\Lambda_{22}=-\Lambda_{11}$ or $\Lambda_{12}=\Lambda_{21}=0$. In the first case $\Lambda$ is traceless, with $\Lambda_{11}^{2}+|\Lambda_{12}|^{2}=1$, i.e. $\Lambda$ can be parametrized as

with $\theta\in[0,2\pi)$ and $\phi\in[0,2\pi)$, and $\det\Lambda=-1$. In the second case $\Lambda$ is diagonal with $\Lambda_{11}^{2}=\Lambda_{22}^{2}=1$. The possibility that $\Lambda_{11}=-\Lambda_{22}$ is already included in the first case, leaving $\Lambda=\pm e$ as the only new possibilities, and for which $\det\Lambda=1$.∎

In more general terms, if a $2\times 2$ unitary matrix $\Lambda$ is selfadjoint, either $\Lambda\subset SU(2)$, or $i\Lambda\subset SU(2)$. In particular, the parametrization for $\Lambda$ given in Lemma 5 is directly related to the standard parametrization of $SU(2)$ matrices.

The matrices $A$ and $B$ of Proposition 4 either have the form (for $\Lambda=e$)

with $\omega=-1/2+i\sqrt{3}/2=\exp(2\pi i/3),$ or otherwise (for $\Lambda\neq\pm e$)

At this point, all unitarity constraints on the matrix $H$ and its submatrices have been accounted for. Note that although the matrices $A$ and $B$ satisfy the unitarity constraints, they will in general not be Hadamard (the modulus of the matrix elements will not be equal to one).

The unimodularity constraints

The additional condition that all elements of $H$ should be of unit modulus can now be imposed.

Let $H$ be an $H_{2}$-reducible Hadamard matrix on the form (2.4), and let $A$ and $B$ be as in Proposition 4 and Corollary 6. For $A$ and $B$ according to (3.4) or (3.5), the elements of $a$, $b$, $c$ and $d$ are of unit modulus if

For $A$ and $B$ according to (3.6), the elements of $a$, $b$, $c$ and $d$ are of unit modulus if

The elements of $a=\frac{1}{2}Z_{3}AZ_{1}$ can all be expressed in terms of $a_{11}(z_{1},z_{3})=(A_{11}+z_{1}A_{12}+z_{3}A_{21}+z_{1}z_{3}A_{22})/2$,

For $A$ and $B$ according to (3.4) or (3.5), the four conditions

all reduce to the first of the relations (4.1), while for $A$ and $B$ according to (3.6), the first of the relations (4.2) is obtained. The remaining relations follow in a similar manner by considering $b$, $c$ and $d$. ∎

With this results, all the conditions needed to characterize the set of $H_{2}$-reducible Hadamard matrices have been given in an explicit form. Before examining these conditions in detail, however, some additional constraints will be imposed that come from the desire to obtain a characterization in terms of inequivalent matrices.

Given $A$ and $B$, the 16 possible sign combinations for the $z_{i}$ parameters obtained when solving (4.1) or (4.2) generate equivalent sets of Hadamard matrices.

The conditions (4.1) and (4.2) only determine the $z_{i}$ parameters up to a sign. However, a sign change can be compensated by an interchange of rows and/or of columns, and the resulting Hadamard matrix is therefore equivalent to the original one. For instance, let $H^{\prime}$ and $H^{\prime\prime}$ only differ in the sign of $z_{3}$,

where P=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right) is a row-permuting matrix. Then

As a consequence of Proposition 8, in order to map out the family of all non-equivalent $H_{2}$-reducible Hadamard matrices, only one sign for the $z_{i}$ parameters needs to be considered.

It can also be shown that without loosing inequivalent matrices the range of the $\theta$ and $\phi$ parameters of Lemma 5 can be reduced to $[0,\pi)$, and the special cases corresponding to $\Lambda=\pm e$ (i.e. to Eqns (3.4) and (3.5)) can be disregarded.

Any $H_{2}$-reducible Hadamard matrix is equivalent to a matrix on the form specified in Proposition 4, with

for $\theta\in[0,\pi)$, $\phi\in[0,\pi)$.

From Proposition 4 it follows that a change of sign $\Lambda\to-\Lambda$ induces the interchange $A\leftrightarrow B$,

When the interchange $A\leftrightarrow B$ is carried out in (4.2), the resulting equations for the $z$-parameters are changed. If, however, the original equations had solutions $z_{1}$, $z_{2}$, $z_{3}$ and $z_{4}$, the new equations will have solutions $z_{1}^{\prime}=z_{1}$, $z_{2}^{\prime}=z_{2}$, $z_{3}^{\prime}=z_{4}$ and $z_{4}^{\prime}=z_{3}$. Therefore,

where in the last step some rows have been permuted. Therefore, in order to map out the family of all non-equivalent $H_{2}$-reducible Hadamard matrices, only one sign for $\Lambda$ needs to be considered.

For the $\Lambda\neq\pm e$ case, the transformations $(\theta,\phi)\to(\theta+\pi,\phi)$ and $(\theta,\phi)\to(\pi-\theta,\phi+\pi)$ both imply $\Lambda\to-\Lambda$. As a result, the range for $\theta$ and $\phi$ can be reduced to $[0,\pi)$ .

For the $\Lambda=\pm e$ case, only $\Lambda=e$ needs to be considered further, and it will first be shown that the resulting Hadamard family is equivalent to either of the two Fourier families. Indeed, from (4.1), either $z_{3}^{2}=z_{4}^{2}=1$, with $z_{1}^{2}$ and $z_{2}^{2}$ unconstrained, or $z_{1}^{2}=z_{2}^{2}=1$, with $z_{3}^{2}$ and $z_{4}^{2}$ unconstrained. In the first case, let $z_{3}=z_{4}=$1, so that $Z_{3}=Z_{4}=F_{2}$. The resulting Hadamard matrices (see Corollary 6)

build the Fourier family, $F_{6}^{(2)}$, with $z_{1}$ and $z_{2}$ as parameters. In the second case, let $z_{1}=z_{2}=$1, so that $Z_{1}=Z_{2}=F_{2}$, and the resulting matrices build the Fourier transposed family ($F_{6}^{(2)})^{T}$, with $z_{3}$ and $z_{4}$ as parameters,

However, as will be seen in the next section, $F_{6}^{(2)}$ and ($F_{6}^{(2)})^{T}$ also appear as limit families in the $\Lambda\neq\pm e$ case, for $\theta\to 0$ and $\theta\to\pi/2$. For the purpose of classifying all $H_{2}$-reducible Hadamard matrices, the $\Lambda=\pm e$ cases can therefore be disregarded from now on. ∎

The three-parameter family

Given the matrices $A$ and $B$ of Proposition 4, or more precisely the parameters $\theta$ and $\phi$ of Proposition 9, what remains is to determine in detail the conditions on the parameters $z_{i}$ that follow from the unimodularity constraints (4.2). It is useful to see these constraints as Möbius transformations

that, as long as $|\alpha|^{2}-|\beta|^{2}\neq 0$, map the unit circle onto itself. Formally, from (4.2),

with $\alpha_{A}=A_{12}^{2}$, $\beta_{A}=A_{11}^{2}$, and $\alpha_{B}=B_{12}^{2}$, $\beta_{B}=B_{11}^{2}$. Recall that the inverse of a Möbius transformation, as well as a sequence of two Möbius transformations, is also a Möbius transformation.

Through straightforward calculation, the following relation between $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ can easily be verified (using the expressions for $A$ and $B$ in terms of the $\Lambda$ of Propositions 4 and 9).

In view of Proposition 10, the relations (5.1) are not independent, but only allow for expressing three of the parameters $z_{i}$ in terms of the fourth. Let for instance $z_{1}=\exp(i\psi_{1})$ where, considering Proposition 9, $\psi_{1}\in[0,\pi)$. Then

and the resulting set of Hadamard matrices will depend on the three parameters $\theta,\,\phi$ and $\psi_{1}$. The same set will be generated starting from any other $z_{i}$, and constitutes the advertised three-parameter family of complex Hadamard matrices of order 6.

The Möbius transformations (5.1) become degenerate if $|\alpha|^{2}-|\beta|^{2}\to 0$: the transformation $w=\mathcal{M}(z)$ degenerates into a mapping of the unit circle in $z$ into a single point $w=\alpha/\bar{\beta}$, and this mapping has no inverse, and the inverse transform $z=\mathcal{M}^{-1}(w)$ degenerates into a mapping of the unit circle in $w$ into a single point $z=\bar{\alpha}/\bar{\beta}$, and again there is no inverse. For $\mathcal{M}_{A}$ and $\mathcal{M}_{A}^{-1}$ this occurs if $|A_{11}|=|A_{12}|=1$, i.e. if

and for $\mathcal{M}_{B}$ and $\mathcal{M}_{B}^{-1}$ if $|B_{11}|=|B_{12}|=1$, i.e. if

Both transformations are degenerate when $\theta=0$, any $\phi$ (and also when $\theta\to\pi$, any $\phi$), and when $\theta=\pi/2$, $\phi=0$ (and also when $\theta=\pi/2$, $\phi\to\pi$).

In general, such a degeneracy does not prevent the construction of the three-parameter family as outlined above (see Appendix 1). However, at the points where both transformations are degenerate, the analysis must take into account that these points can be reached not only along the degeneracy curves but from an arbitrary direction in the $\theta-\phi$ plane. The resulting limit families may be obtained either through an explicit limiting procedure, as exemplified in Appendix 2, or in the following direct manner.

If $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ are both degenerate, there are two cases to be considered. First, if $\theta=0$ then \Lambda=\left(\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right) for any $\phi$, so that

Here, \Omega=\left(\begin{array}[]{cc}\omega&0\\ 0&\omega^{2}\end{array}\right) with $1+\omega+\omega^{2}=0$ and $e+\Omega+\Omega^{2}=0$ (recall that $\omega=\exp(2\pi i/3)$). The unimodularity conditions (4.2) take the form

This set requires that $z_{3}^{2}=1$ and/or $z_{4}^{2}=1$. If $z_{3}^{2}=z_{4}^{2}=1$, then there are no restrictions on $z_{1}$ or $z_{2}$. Since all sign combinations result in equivalent Hadamard matrices, let $z_{3}=z_{4}=1$. Then $Z_{3}=Z_{4}=F_{2}$, and the resulting Hadamard family is equivalent to the Fourier family $F_{6}^{(2)}$,

The system (5.2) is also satisfied if $z_{3}^{2}=1$, $z_{1}^{2}=\omega^{2}$ and $z_{2}^{2}=\omega^{4}$, with $z_{4}$ arbitrary. Let $z_{3}=1,$ $z_{1}=\omega$ and $z_{2}=\omega^{2}$. In this case

and this family is equivalent to a subfamily of $(F_{6}^{(2)})^{T}$. Finally, the system (5.2) is also satisfied if $z_{4}^{2}=1$, $z_{1}^{2}=\omega^{4}$ and $z_{2}^{2}=\omega^{2}$, with $z_{3}$ arbitrary. Like in the previous case, the resulting $H$ can be shown to be equivalent to a subfamily of $(F_{6}^{(2)})^{T}$.

The Möbius transformations $\mathcal{M}_{A}$ and $\mathcal{M}_{B}$ are also degenerate when $\theta=\pi/2$, $\phi=0$, and in this case \Lambda=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right) and

The subsequent analysis is similar to the previous one, and results in matrix families that either are equivalent to $(F_{6}^{(2)})^{T}$, or to one-parameter subfamilies of $F_{6}^{(2)}$.

The finding of two-parameter subfamilies at the doubly degenerate points might not have been expected, since two ($\theta$ and $\phi$) of the three original parameters have been eliminated. It might be recalled, however, that a similar phenomenon was observed in , where the two-parameter family $K_{6}^{(2)}$ at certain fixed parameter values had the one-parameter $D_{6}^{(1)}$ family as limit family. As was detailed in , the extra parameter enters since the limit family depends on the direction from which the limit point is reached, just as is observed here (see Appendix 2).

It should be recalled that the appearance of the Fourier and Fourier transposed families in the present context was made use of in the proof of Proposition 9.

With these observations, the classification problem for $H_{2}$-reducible Hadamard matrices is solved. The main results of the present paper are collected in the following theorem.

Any $H_{2}$-reducible (complex) Hadamard matrices (of order 6) is equivalent to a member of the three-parameter family of dephased matrices

Here F_{2}=\left(\begin{array}[]{cc}1&1\\ 1&-1\end{array}\right), $A$ is the matrix

for any $\theta\in[0,\pi)$ and $\phi\in[0,\pi)$, and $B=-F_{2}-A$. In the matrices Z_{i}=\left(\begin{array}[]{cc}1&1\\ z_{i}&-z_{i}\end{array}\right), $i=1,2$, and Z_{i}=\left(\begin{array}[]{cc}1&z_{i}\\ 1&-z_{i}\end{array}\right), $i=3,4$, the parameters $z_{i}$ are related through Möbius transformations

with $\alpha_{A}=A_{12}^{2}$, $\beta_{A}=A_{11}^{2}$, and $\alpha_{B}=B_{12}^{2}$, $\beta_{B}=B_{11}^{2}$. In general, one of the parameters $z_{i}$ can be chosen freely, say $z_{1}=\exp(i\psi_{1})$, $\psi_{1}\in[0,\pi)$, in which case $z_{2}^{2}=\mathcal{M}_{A}^{-1}(\mathcal{M}_{B}(z_{1}^{2}))=\mathcal{M}_{B}^{-1}(\mathcal{M}_{A}(z_{1}^{2}))$, $z_{3}^{2}=\mathcal{M}_{A}(z_{1}^{2})$ and $z_{4}^{2}=\mathcal{M}_{B}(z_{1}^{2})$. Any sign combinations for $z_{1}$, $z_{2}$, $z_{3}$ and $z_{4}$ lead to three-parameter families that are equivalent to each other.

Special cases

As pointed out above, all so far (analytically) known one- and two-parameter families of complex Hadamard matrices of order 6 are subfamilies of the three-parameter family constructed in the previous sections. In general, however, the parameters used to classify these subfamilies differ from the parameters introduced here, and the detailed connection is not always transparent. For instance, the two-parameter family $K_{6}^{(2)}$ of exploits simplifications entailed by the assumption that $z_{2}=z_{1}$ and $z_{4}=z_{3}$. Such an assumption is less natural from the point of view of the parametrization developed in the present paper, and amounts to introducing a dependence between $z_{1}$ and the parameters $\theta$ and $\phi$. In this respect, the family $D_{6}^{(1)}$ is an exception, as will be shown next.

Particularly simple subfamilies of the three-parameter family can be expected if $\theta$ and $\phi$ kept constant. Consider for example the point $\theta=\arccos(1/\sqrt{3})$, $\phi=\pi/4$, for which

Since $|A_{11}|=|A_{12}|=1$, $\mathcal{M}_{A}$ is degenerate and $\mathcal{M}_{A}(z^{2})=\mathcal{M}_{A}^{-1}(z^{2})=-1$. Furthermore, $\mathcal{M}_{B}(z^{2})=\mathcal{M}_{B}^{-1}(z^{2})=1/z^{2}=\bar{z}^{2}$ so that, taking $z_{1}=z$ as independent parameter, $z_{2}^{2}=-1$, $z_{3}^{2}=-1$ and $z_{4}^{2}=\bar{z}^{2}$. Let $z_{2}=z_{3}=i$ and $z_{4}=\bar{z}$. The resulting one-parameter Hadamard matrix

is equivalent to the generic member of $D_{6}^{(1)}$.

Another example of a simple subfamily can be obtained as follows. For points on the $\mathcal{M}_{A}$ degeneracy curve, $\sin\phi=\sqrt{3}\cos\theta\cos\phi$ (see Section 5). Along this curve $A_{12}=\exp(2i\phi)A_{11}$ and $\mathcal{M}_{A}^{-1}(z^{2})=\exp(-4i\phi)$, all $z^{2}$. Let $z_{1}=z=\exp(i\psi)$. If $\phi$ is chosen equal to $-\psi/2$ then $z_{2}^{2}=\mathcal{M}_{A}^{-1}(z^{2})=z^{2}$, i.e. $z_{1}=z_{2}=z$. Furthermore

for $\psi\in[0,2\pi/3]$, and the resulting Hadamard matrix is equivalent to

This one-parameter Hadamard family can be identified as a subfamily of $K_{6}^{(2)}$.

Summary and outlook

With the results of the present paper, the characterization problem for complex Hadamard matrices of order six has been given a partial solution, in that the subset of $H_{2}$-reducible Hadamard matrices has been fully described in terms of a single, three-parameter family. There is strong numerical evidence, based on some $10^{5}$ non-reducible Hadamard matrices (see also ) that a full characterization requires an additional parameter, but it remains an open question whether or not closed form expressions for such a four-parameter family can be found.

The parameters chosen here for the three-parameter family are not unique, but appear as a natural choice. Minor variations, like choosing the (SU(2)) parameters $\theta$ and $\phi$ differently, offer no obvious advantage.

As an application of the results presented here, Hadamard matrices in 12 dimensions can be constructed. Such an extension was outlined in based on the at the time known two-parameter families in six dimensions. A corresponding extension based on the three-parameter family of the present paper results in an eleven-parameter family, the largest family constructed so far in 12 dimensions.

Appendix 1: Degenerate transformations

If one of the Möbius transformations (5.1) becomes degenerate, the three-parameter family may still be constructed as outlined in Section 5, but the result may depend on how the degeneracy limit is approached. In order to illustrate this point, let $\mathcal{M}_{A}$ but not $\mathcal{M}_{B}$ be degenerate. In such a case, $\mathcal{M}_{A}(z^{2})=w_{0}^{2}$ and $\mathcal{M}_{A}^{-1}(z^{2})=z_{0}^{2}$ for any $z$, where $w_{0}^{2}=\alpha_{A}/\bar{\beta}_{A}$ and $z_{0}^{2}=\bar{\alpha}_{A}/\bar{\beta}_{A}$ are uniquely specified by $\theta$ or $\phi$ along the degeneracy curve. Furthermore, $\mathcal{M}_{B}(z_{0}^{2})=w_{0}^{2}$ as a consequence of Proposition 10. Using $z_{1}$ or $z_{4}$ as independent parameter, the remaining parameters are obtained through $z_{4}^{2}=\mathcal{M}_{B}(z_{1}^{2})$ with $z_{3}^{2}=w_{0}^{2}$ and $z_{2}^{2}=z_{0}^{2}$. On the other hand, taking $z_{2}$ or $z_{3}$ as independent parameter leads to $z_{3}^{2}=\mathcal{M}_{B}(z_{2}^{2})$ with $z_{1}^{2}=z_{0}^{2}$ and $z_{4}^{2}=w_{0}^{2}$. The resulting two limit matrices,

in obvious notation, are seemingly different, but an interchange of rows and of columns shows that they generate families of equivalent Hadamard matrices. There is therefore no need to amend the general construction in Section 5 with additional rules when one of the Möbius transformations becomes degenerate.

Appendix 2: The limit families at θ=0𝜃0\theta=0

In order to see how the general, three-parameter family behaves when the doubly degenerate point at $\theta=0$ is approached, let $\theta$ be infinitesimal in the expressions for $A$ and $B$ in Theorem 11,

where $\omega=\exp(2\pi i/3)$. The coefficients of the Möbius transformations (5.1) are

Choosing $z_{1}$ as the independent parameter results in the relations, when $\theta\to 0$,

Given $z_{1}$ (not equal to $\omega$ or $\omega^{2}$), $\phi$ maps out a unit circle in $z_{2}$, i.e. the Fourier family $F_{6}^{(2)}$ with $z_{1}$ and $z_{2}$ as independent parameters is obtained.

On the other hand, choosing $z_{3}$ as independent parameter results in

Therefore, any $z_{3}\neq 1$ leaves the other three parameters fixed, and, as detailed in section 6.3, the resulting Hadamard family is equivalent to a subfamily of ($F_{6}^{(2)})^{T}$.

References