Catalog of CHM

Appendix C: Symmetric and Hermitian CHM

created: 2023-01-14

updated: 2024-07-14 by W. Bruzda

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 roughly described symbolically, it is subjected to a random walk procedure to restore the original numerical values, or – as it is in many cases – to discover other independent solutions provided by the particular pattern. So, it looks like having a numerical matrix representation, we are trying a different method on gently improved internal structure to confirm the result... and... it works perfectly! Here, we decided not to use pure algebra to solve the matrix elements via unitarity constraints (still too few information about the matrix), rather, we rely only on the numerical optimizing a certain objective function, which proved to be a moderately good method for such an elementary research, and even this allows us to present many examples analytically.

Obviously, the convergence deteriorates with increasing dimension N, and the average time of obtaining a solution takes longer and longer. Some matrices or families might be parts of more general objects. Still, many new solutions can be obtained by fixing particular entries. Also, many different patterns might share the same set of solutions, and surely not all possible solutions have been encountered, so the work is actually under construction. The list shall be updated constantly until a tool appears that solves the classification problem definitely (which means, we have a lot of time...).

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 symmetrizable and isolated.

N > 11

Hermitian CHM (including families)↑

One can easily prove that Hermitian CHM can exist only in even dimensions N ∈ 2N. No explicitly isolated example was found so far...

N = 6

N = 8

N = 10

N = 12

N > 12

Unresolved Cases↑

Not every (complex) Hadamard matrix can be (easily) brought to its symmetric equivalent. Soon, the list of such matrices will be printed here... If anybody knows such complex Hadamard examples, please contact us.