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Complex Hadamard Matrices and Applications Teo Banica Complex Hadamard matrices and applications Teo Banica To cite this version: Teo Banica. Complex Hadamard matrices and applications. 2021. hal-02317067v2 HAL Id: hal-02317067 https://hal.archives-ouvertes.fr/hal-02317067v2 Preprint submitted on 27 Jul 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. COMPLEX HADAMARD MATRICES AND APPLICATIONS TEO BANICA Abstract. A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries are on the unit circle, jHijj = 1, and whose rows and pairwise orthogonal. The main ij 2πi=N example is the Fourier matrix, FN = (w ) with w = e . We discuss here the basic theory of such matrices, with emphasis on geometric and analytic aspects. Contents Introduction2 1. Hadamard matrices 7 2. Analytic aspects 25 3. Norm maximizers 43 4. Partial matrices 61 5. Complex matrices 79 6. Roots of unity 97 7. Geometry, defect 115 8. Special matrices 133 9. Circulant matrices 151 10. Bistochastic form 169 11. Glow computations 187 12. Local estimates 205 13. Quantum groups 223 14. Hadamard models 241 15. Generalizations 259 16. Fourier models 277 References 295 2010 Mathematics Subject Classification. 15B10. Key words and phrases. Hadamard matrix, Fourier matrix. 1 2 TEO BANICA Introduction A complex Hadamard matrix is a square matrix H 2 MN (C) whose entries are the unit circle, jHijj = 1, and whose rows are pairwise orthogonal with respect to the usual scalar product of CN . These matrices appear in connection with many questions, the basic example being the Fourier matrix, which is as follows, with w = e2πi=N : ij FN = (w )ij In standard matrix form, and with the standard discrete Fourier analysis convention that the indices vary as i; j = 0; 1;:::;N − 1, this matrix is as follows: 01 1 1 ::: 1 1 B1 w w2 : : : wN−1 C B C B1 w2 w4 : : : w2(N−1)C FN = B C B. C @. A 1 wN−1 w2(N−1) : : : w(N−1)2 Observe that this matrix is indeed Hadamard, with the orthogonality formulae between rows coming from the fact that the barycenter of any centered regular polygon is 0. This matrix is the matrix of the discrete Fourier transform over ZN , and the arbitrary complex Hadamard matrices can be thought of as being \generalized Fourier matrices". As an illustration, the complex Hadamard matrices cover in fact all the classical discrete Fourier transforms. Consider indeed the Fourier coupling of an arbitrary finite abelian group G, regarded via the isomorphism G ' Gb as a square matrix, FG 2 MG(C): F =< i; j > G i2G;j2Gb For the cyclic group G = ZN we obtain in this way the above matrix FN . In gen- eral, we can write G = ZN1 × ::: × ZNk , and modulo some standard identifications, the corresponding Fourier matrix decomposes then over components, as follows: FG = FN1 ⊗ ::: ⊗ FNk Now since the tensor product of complex Hadamard matrices is Hadamard, we conclude that this generalized Fourier matrix FG is a complex Hadamard matrix. The above generalization is particularly interesting when taking as input the groups n G = Z2 . Indeed, for such groups the corresponding Fourier matrices are real. As a first example here, at n = 1 we obtain the Fourier matrix F2, which is as follows: 1 1 F = 2 1 −1 HADAMARD MATRICES 3 This matrix is also denoted W2, and called first Walsh matrix. At n = 2 now, we obtain the second Walsh matrix, W4 = W2 ⊗ W2, which is given by: 01 1 1 1 1 B1 −1 1 −1C W4 = B C @1 1 −1 −1A 1 −1 −1 1 ⊗n In general, we obtain in this way the tensor powers W2n = F2 of the Fourier matrix n F2, having size N = 2 , called Walsh matrices, and having many applications. The world of the real Hadamard matrices, H 2 MN (±1), is quite fascinating, hav- ing deep ties with combinatorics, coding, and design theory. Due to the orthogonality conditions between the first 3 rows, the size of such a matrix must satisfy: N 2 f2g [ 4N The celebrated Hadamard Conjecture (HC), which is more than 100 years old, and is one of the most beautiful problems in combinatorics, and mathematics in general, states that real Hadamard matrices should exist at any N 2 4N. Famous as well is the Circulant Hadamard Conjecture (CHC), which is more than 50 years old, stating that the following \conjugate" of the above matrix W4, and its other conjugates, are the unique circulant real Hadamard matrices, and this regardless of the value of N 2 N: 0−1 1 1 1 1 B 1 −1 1 1 C K4 = B C @ 1 1 −1 1 A 1 1 1 −1 There are many other interesting questions regarding the real Hadamard matrices, mostly of algebraic nature, but with some subtle analytic aspects as well. Getting back now to the general complex case, H 2 MN (T), the situation here is quite different. We know that at any N 2 N we have the Fourier matrix FN , so the HC dissapears. The CHC dissapears in fact as well, because the Fourier matrix FN can be put in circulant form, up to some simple operations on the rows and columns. As an 2πi=3 example, consider the Fourier matrix F3, which is as follows, with w = e : 01 1 1 1 2 F3 = @1 w w A 1 w2 w By doing some suitable manipulations on the rows and columns, namely permuting them, or multiplying them by numbers w 2 T, which are operations which preserve the 4 TEO BANICA class of complex Hadamard matrices, we can put F3 in circulant form, as follows: 0w 1 11 0 F3 = @1 w 1A 1 1 w The situation at general N 2 N is similar, and as a conclusion, both the HC and CHC dissapear, in the general complex setting H 2 MN (T). It is possible however to recover these conjectures, in a more complicated form, by looking at the Hadamard matrices H 2 MN (Zs) having as entries roots of unity of a given order s 2 N. In the purely complex case, where H 2 MN (T) is allowed to have non-roots of unity as entries, the whole subject rather belongs to geometry. Indeed, for a matrix H 2 MN p(T), the orthogonality condition between the rows tells us that the rescaled matrix U = H= N must be unitary. Thus, the N × N complex Hadamard matrices form a real algebraic manifold, appearing as an intersection of smooth manifolds, as follows: p XN = MN (T) \ NUN This intersection is far from being smooth, and generally speaking, the study of the complex Hadamard matrices belongs to real algebraic geometry, and more specifically to a real algebraic geometry of rather arithmetic type, related to Fourier analysis. As an illustration for the various phenomena that might appear, let us briefly discuss now the classification of the complex Hadamard matrices, at small values of N 2 N. At N = 2; 3 the situation is elementary, with F2;F3 being the unique matrices, up to equivalence. At N = 4 now, the solutions are the affine deformations of the Walsh matrix W4, depending on complex number of modulus one q 2 T, as follows: 01 1 1 1 1 q B1 −1 1 −1C W = B C 4 @1 q −1 −qA 1 −q −1 q At N = 5 now, the situation becomes quite complicated, but Haagerup was able to prove in [69] that, up to equivalence as usual, the unique complex Hadamard matrix is 2πi=5 the Fourier matrix F5, which is as follows, with w = e : 01 1 1 1 1 1 B1 w w2 w3 w4C B 2 4 3C F5 = B1 w w w w C B 3 4 2C @1 w w w w A 1 w4 w3 w2 w At N = 6 the complex Hadamard matrices are not classified yet, but there are many known interesting examples. First, since 6 = 2 × 3 is a composite number, the Fourier HADAMARD MATRICES 5 matrix F6 admits certain affine deformations, which are well understood. Next, we have the following matrix from [69], depending on a parameter on the unit circle, q 2 T: 01 1 1 1 1 1 1 B1 −1 i i −i −iC B C q B1 i −1 −i q −qC H6 = B C B1 i −i −1 −q q C @1 −i q¯ −q¯ i −1A 1 −i −q¯ q¯ −1 i We have as well the following matrix discovered and used by Tao in [131], with w = e2πi=3, which has the property of being \isolated", in an appropriate sense: 01 1 1 1 1 1 1 2 2 B1 1 w w w w C B 2 2 C B1 w 1 w w w C T6 = B 2 2C B1 w w 1 w w C @1 w2 w2 w 1 w A 1 w2 w w2 w 1 As explained in [17], up to the Hadamard equivalence relation, the above-mentioned matrices are the only ones at N = 6, whose combinatorics is based on the roots of unity. Finally, at N = 7, even the matrices whose combinatorics is based on the roots of unity are not classified yet.
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