About: Complex Hadamard matrix

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A complex Hadamard matrix is any complex matrix satisfying two conditions: * unimodularity (the modulus of each entry is unity): * orthogonality: , where denotes the Hermitian transpose of and is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix can be made into a unitary matrix by multiplying it by ; conversely, any unitary matrix whose entries all have modulus becomes a complex Hadamard upon multiplication by . belong to this class.

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  • A complex Hadamard matrix is any complex matrix satisfying two conditions: * unimodularity (the modulus of each entry is unity): * orthogonality: , where denotes the Hermitian transpose of and is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix can be made into a unitary matrix by multiplying it by ; conversely, any unitary matrix whose entries all have modulus becomes a complex Hadamard upon multiplication by . 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 (compare the real case, in which existence is not known for every ). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor), belong to this class. (en)
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  • A complex Hadamard matrix is any complex matrix satisfying two conditions: * unimodularity (the modulus of each entry is unity): * orthogonality: , where denotes the Hermitian transpose of and is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix can be made into a unitary matrix by multiplying it by ; conversely, any unitary matrix whose entries all have modulus becomes a complex Hadamard upon multiplication by . belong to this class. (en)
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  • Complex Hadamard matrix (en)
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