In mathematics, a complex Hadamard matrixH of size N with all its columns (rows) mutually orthogonal, belongs to the Butson-typeH(q, N) if all its elements are powers of q-th root of unity,
If p is prime and , then can exist only for with integer m and it is conjectured they exist for all such cases with . For , the corresponding conjecture is existence for all multiples of 4. In general, the problem of finding all sets such that the Butson - type matrices exist, remains open.
A. T. Butson, Generalized Hadamard matrices, Proc. Am. Math. Soc. 13, 894-898 (1962).
A. T. Butson, Relations among generalized Hadamard matrices, relative difference sets, and maximal length linear recurring sequences, Can. J. Math. 15, 42-48 (1963).
R. J. Turyn, Complex Hadamard matrices, pp. 435–437 in Combinatorial Structures and their Applications, Gordon and Breach, London (1970).
External linksEdit
Complex Hadamard Matrices of Butson type - a catalogue, by Wojciech Bruzda, Wojciech Tadej and Karol Życzkowski, retrieved October 24, 2006
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butson, type, hadamard, matrix, mathematics, complex, hadamard, matrix, size, with, columns, rows, mutually, orthogonal, belongs, butson, type, elements, powers, root, unity, displaystyle, quad, quad, dots, contents, existence, examples, references, external, . In mathematics a complex Hadamard matrix H of size N with all its columns rows mutually orthogonal belongs to the Butson type H q N if all its elements are powers of q th root of unity H j k q 1 f o r j k 1 2 N displaystyle H jk q 1 quad rm for quad j k 1 2 dots N dd Contents 1 Existence 2 Examples 3 References 4 External linksExistence EditIf p is prime and N gt 1 displaystyle N gt 1 then H p N displaystyle H p N can exist only for N m p displaystyle N mp with integer m and it is conjectured they exist for all such cases with p 3 displaystyle p geq 3 For p 2 displaystyle p 2 the corresponding conjecture is existence for all multiples of 4 In general the problem of finding all sets q N displaystyle q N such that the Butson type matrices H q N displaystyle H q N exist remains open Examples EditH 2 N displaystyle H 2 N contains real Hadamard matrices of size N H 4 N displaystyle H 4 N contains Hadamard matrices composed of 1 i displaystyle pm 1 pm i such matrices were called by Turyn complex Hadamard matrices in the limit q displaystyle q to infty one can approximate all complex Hadamard matrices Fourier matrices F N j k exp 2 p i j 1 k 1 N f o r j k 1 2 N displaystyle F N jk exp 2 pi i j 1 k 1 N quad rm for quad j k 1 2 dots N belong to the Butson type F N H N N displaystyle F N in H N N dd whileF N F N H N N 2 displaystyle F N otimes F N in H N N 2 dd F N F N F N H N N 3 displaystyle F N otimes F N otimes F N in H N N 3 dd D 6 1 1 1 1 1 1 1 1 i i i i 1 i 1 i i i 1 i i 1 i i 1 i i i 1 i 1 i i i i 1 H 4 6 displaystyle D 6 begin bmatrix 1 amp 1 amp 1 amp 1 amp 1 amp 1 1 amp 1 amp i amp i amp i amp i 1 amp i amp 1 amp i amp i amp i 1 amp i amp i amp 1 amp i amp i 1 amp i amp i amp i amp 1 amp i 1 amp i amp i amp i amp i amp 1 end bmatrix in H 4 6 dd S 6 1 1 1 1 1 1 1 1 z z z 2 z 2 1 z 1 z 2 z 2 z 1 z z 2 1 z z 2 1 z 2 z 2 z 1 z 1 z 2 z z 2 z 1 H 3 6 displaystyle S 6 begin bmatrix 1 amp 1 amp 1 amp 1 amp 1 amp 1 1 amp 1 amp z amp z amp z 2 amp z 2 1 amp z amp 1 amp z 2 amp z 2 amp z 1 amp z amp z 2 amp 1 amp z amp z 2 1 amp z 2 amp z 2 amp z amp 1 amp z 1 amp z 2 amp z amp z 2 amp z amp 1 end bmatrix in H 3 6 where z exp 2 p i 3 displaystyle z exp 2 pi i 3 dd References EditA T Butson Generalized Hadamard matrices Proc Am Math Soc 13 894 898 1962 A T Butson Relations among generalized Hadamard matrices relative difference sets and maximal length linear recurring sequences Can J Math 15 42 48 1963 R J Turyn Complex Hadamard matrices pp 435 437 in Combinatorial Structures and their Applications Gordon and Breach London 1970 External links EditComplex Hadamard Matrices of Butson type a catalogue by Wojciech Bruzda Wojciech Tadej and Karol Zyczkowski retrieved October 24 2006 This article about matrices is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Butson type Hadamard matrix amp oldid 1170016207, wikipedia, wiki, book, books, library,