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Article Contents

# Informing the structure of complex Hadamard matrix spaces using a flow

• A complex Hadamard matrix $H$ may be isolated or may lie in a higher-dimensional space of Hadamards. We provide an upper bound for this dimension as the dimension of the center subspace of a gradient flow and apply the Center Manifold Theorem of dynamical systems theory to study local structure in spaces of complex Hadamard matrices. Through examples, we provide several applications of our methodology including the construction of affine families of Hadamard matrices.

Mathematics Subject Classification: Primary: 37C10, 05B20; Secondary: 37L10.

 Citation:

• Figure 1.  Plot of the eigenvalues of the linearization of $\Phi_4$ at ${\boldsymbol \theta}(a)$ for $a \in [0,\pi]$. $\lambda_1(a)$ (blue), $\lambda_3(a)$ (green), and $\lambda_6(a)$ (red) simultaneously vanish at $a = \pi/2$, while all other eigenvalues (gray) are strictly negative for all $a \in [0,\pi]$

Figure 2.  Snapshots of 500 initial phases, drawn from $\mathbb{R}^{9}$ uniformly at random from a neighborhood of the core phases corresponding to $F(\pi/2)$, as they evolve under the flow defined by $\Phi_4({\boldsymbol \theta})$, at times (ⅰ) 5, (ⅱ) 20, (ⅲ) 70, and (ⅳ) 500. Each point cloud has been projected onto its top three principal components, and each point ${\boldsymbol \theta}$ is colored by $\log_{10}$ of the magnitude of the vector field $\Phi_4({\boldsymbol \theta})$

Figure 3.  Plot of the 25 eigenvalues of $D\Phi_6\vert_{D_6(c)}$ for $c \in [-\pi/2,\pi/2]$. The zero eigenvalue (blue) has multiplicity four, the roots of $f_1(\lambda;c)$ (green) have multiplicity two, and all other eigenvalues (gray) are simple

Figure 4.  Snapshots of 500 initial phases, drawn from $\mathbb{R}^{64}$ uniformly at random from a neighborhood of the core phases corresponding to $B_9^{(0)}$, as they evolve under the flow defined by $\Phi_9({\boldsymbol \theta})$, at times (ⅰ) 5, (ⅱ) 20, (ⅲ) 70, and (ⅳ, ⅴ) 500. Each point cloud has been projected onto its top three principal components, and each point ${\boldsymbol \theta}$ is colored by $\log_{10}$ of the magnitude of the vector field $\Phi_9({\boldsymbol \theta})$

Table 1.  Nonzero coordinates of the vectors in the basis $\{\textbf{V}_1, \ldots, \textbf{V}_{16}\}$ for $D\Phi_{10}\vert_{D_{10}}$. A nonzero coordinate has value 1 or -1, indicated by the subcolumn to which it belongs

 1 -1 vector coordinate $\textbf{V}_1$ 2 3 7 8 74 75 79 80 10 18 19 27 55 63 64 72 $\textbf{V}_2$ 10 12 16 18 64 66 70 72 2 8 20 26 56 62 74 80 $\textbf{V}_3$ 28 29 35 36 46 47 53 54 4 6 13 15 67 69 76 78 $\textbf{V}_4$ 4 8 24 25 40 44 78 79 28 32 48 54 57 63 64 68 $\textbf{V}_5$ 37 40 47 54 55 58 65 72 5 7 15 17 32 34 78 80 $\textbf{V}_6$ 4 9 12 14 48 50 67 72 20 24 28 35 38 42 73 80 $\textbf{V}_7$ 19 27 29 34 38 43 64 72 3 8 13 14 58 59 75 80 $\textbf{V}_8$ 37 39 43 45 46 48 52 54 5 6 23 24 59 60 77 78 $\textbf{V}_9$ 2 8 20 26 43 45 52 54 10 12 59 60 64 66 77 78 $\textbf{V}_{10}$ 47 48 49 50 74 75 76 77 15 18 24 27 33 36 42 45 $\textbf{V}_{11}$ 2 6 30 32 65 69 75 77 10 17 22 27 40 45 46 53 $\textbf{V}_{12}$ 12 14 30 32 48 50 75 77 20 22 24 27 38 40 42 45 $\textbf{V}_{13}$ 47 50 56 59 65 68 74 77 15 16 17 18 42 43 44 45 $\textbf{V}_{14}$ 25 26 34 35 47 50 74 77 15 18 42 45 57 58 66 67 $\textbf{V}_{15}$ 19 23 28 32 64 68 73 77 3 4 8 9 39 40 44 45 $\textbf{V}_{16}$ 19 23 49 53 58 62 73 77 3 9 33 34 39 45 69 70
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