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Appendix C: Symmetric and Hermitian CHM

created: 2023-01-14

updated: 2026-09-13 by W. Bruzda

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Slightly modified Sinkhorn's algorithm can be used to explore symmetric or Hermitian CHM providing new results almost immediately. It is enough to require that during the iterative procedure one additional step takes the form: X → (X + X^ξ)/2, where ξ = T or ξ = † for the symmetric or Hermitian case, respectively.

Outputs from the Sinkhorn procedure were examined for simple dependencies between entries like in Appendix B, from which we also adapted the system of names reflecting the matrix type.

Once a matrix is symbolically characterized, a random walk procedure is applied to recover its original numerical values, or, in many cases, to discover other independent solutions consistent with the given pattern. In other words, starting from a numerical matrix representation, we explore an alternative, slightly refined approach to the internal structure in order to confirm the result. It works!

We do not rely on pure algebra to determine the matrix elements via unitarity constraints (the available information about the matrix is still insufficient). Instead, we depend on numerical optimization of a suitable objective function, which proved to be a quite effective method for such a preliminary investigation. It even allows to present several examples analytically.

The convergence deteriorates as the dimension N increases and the average time required to obtain a solution grows accordingly. Some matrices or families of matrices may belong to more general structures and many new solutions can still be found by fixing particular entries. Moreover, distinct patterns may share the same set of solutions and we have certainly not encountered all possible solutions yet.

symmetric CHM (including families)↑

Symmetric CHM exist in any N and the canonical example is the Fourier matrix. However, given N, not every equivalence class contains a symmetrizable matrix. In contrast with Hermitian examples, many of them are isolated. Maximal possible number of #Λ(SH_N,d,#Λ) = 1 + τ(N) + τ²(N), where τ(N) = N*(N - 1)/2, is denoted in bold and, as shown below, this limit was not achieved (yet?) in every dimension.

N = 6

N = 7

All (known) CHM of order N = 7 are symmetrizable. Moreover, some of them can have their cores be written in a doubly symmetric form:

P7 in BH(7, 6) =           F7 in BH(7, 7) =
   . . . . . . .              . . . . . . .
   . 5 2 3 5 3 1              . 2 6 3 4 1 5
   . 2 5 3 5 1 3              . 6 4 2 5 3 1
   . 3 3 2 1 5 5              . 3 2 1 6 5 4
   . 5 5 1 2 3 3              . 4 5 6 1 2 3
   . 3 1 5 3 5 2              . 1 3 5 2 4 6
   . 1 3 5 3 2 5              . 5 1 4 3 6 2

Q7 = 
   . . . . . . .
   . a b c d e f with a = 0.888409423250061
   . b c f a d e      b = 0.409693609117209
   . c f e b a d      c = 0.357945512003075
   . d a b e f c      d = 0.805644956713980
   . e d a f c b      e = 0.242507857976937
   . f e d c b a      f = 0.630339632445252

C7C = 
   . . . . . . .
   . a b c d e f with a = 0.627180715553348 (some phases depend on the others)
   . b g h j k e      b = 0.097739974494601
   . c h m n j d      c = 0.011089605651288
   . d j n m h c      d = 0.313590357776674
   . e k j h g b      e = 0.616091109902060
   . f e d c b a      f = 0.529440741058746
                      g = 0.195479948989203
                      h = 0.579388839087143
                      j = 0.795239222369215
                      k = 0.400240726619988
                      m = 0.590478444738430
                      n = 0.276888086961757

C7D = 
   . . . . . . .
   . a b c d e f with a = 0.372819284446652 (some phases depend on the others)
   . b g h j k e      b = 0.902260025505399
   . c h m n j d      c = 0.988910394348712
   . d j n m h c      d = 0.686409642223326
   . e k j h g b      e = 0.383908890097940
   . f e d c b a      f = 0.470559258941254
                      g = 0.804520051010797
                      h = 0.420611160912857
                      j = 0.204760777630785
                      k = 0.599759273380013
                      m = 0.409521555261570
                      n = 0.723111913038243

N = 8

N = 9

N = 10

N = 11

All (known) CHM of order N = 11 are isolated.

N = 12

N = 13

N = 14

N = 15

N = 16

N = 17

N = 18

N = 19

N = 20

N > 20

Hermitian CHM (including families)↑

N = 6

N = 8

N = 9

N = 10

N = 12

N = 14

N = 16

N = 20

N > 20