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Butson-type Hadamard Matrices
In mathematics, a complex Hadamard matrix ''H'' of size ''N'' with all its columns (rows) mutually orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ..., belongs to the Butson-type ''H''(''q'', ''N'') if all its elements are powers of ''q''-th root of unity, :: (H_)^q=1 j,k=1,2,\dots,N. Existence If ''p'' is prime and N>1, then H(p,N) can exist only for N = mp with integer ''m'' and it is conjectured they exist for all such cases with p \ge 3. For p=2, the corresponding conjecture is existence for all multiples of 4. In general, the problem of finding all sets \ such that the Butson - type matrices H(q,N) exist, remains open. Examples *H(2,N) contains real Hadamard matrices of size ''N'', *H(4,N) contains Hadamard matrices composed of \pm 1, \pm i - such m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Matrix
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional parallelotope spanned by the rows of an ''n''×''n'' Hadamard matrix has the maximum possible ''n''-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Matrices
In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows. The ''n''-dimensional parallelotope spanned by the rows of an ''n''×''n'' Hadamard matrix has the maximum possible ''n''-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Hadamard Matrices
A complex Hadamard matrix is any complex N \times N matrix H satisfying two conditions: *unimodularity (the modulus of each entry is unity): , H_, =1 j,k=1,2,\dots,N *orthogonality: HH^ = NI , where denotes the Hermitian transpose of H and I is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix H can be made into a unitary matrix by multiplying it by \frac; conversely, any unitary matrix whose entries all have modulus \frac becomes a complex Hadamard upon multiplication by \sqrt. Complex Hadamard matrices arise in the study of operator algebras and the theory of quantum computation. Real Hadamard matrices and Butson-type Hadamard matrices form particular cases of complex Hadamard matrices. Complex Hadamard matrices exist for any natural N (compare the real case, in which existence is not known for every N). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
