A Comparison of Trace and Spectra of Dephased Hadamard Matrices

Indianapolis, IN

A complex Hadamard matrix is an nxn matrix H whose entries consist of complex numbers of unit modulus and that satisfies the following equation: the matrix formed by transposing H and changing all entries to their complex conjugates multiplied by H is equal to H multiplied by H transposed. A Hadamard matrix is considered to be in dephased form if the first row and first column consist entirely of 1’s. For dephased Hadamard matrices up to size 5x5, it is known that having the same trace implies being spectrally equivalent. But, interestingly, there is a 12x12 example where this does not hold. So, our objective is to discover a pair of 6x6 dephased Hadamard matrices with the same trace but different eigenvalues. Through our research, we see that the 6x6 Fourier matrix, the matrix generated by taking the Kronecker product of the 2x2 and 3x3 Fourier matrices, and the matrix generated by taking the Kronecker product of the 2x2 Fourier matrix and a permutation of the 3x3 Fourier matrix all have the same eigenvalues. But, excitingly, we recognize the Cyclic 6-Roots matrix has different eigenvalues by the fact that they are purely real. This demonstrates that for 6x6 dephased Hadamard matrices, equal trace does not directly imply spectral equivalence.