Complex Hadamard matrix

A complex Hadamard matrix is any complex N × N {displaystyle N imes N} matrix H {displaystyle H} satisfying two conditions: A complex Hadamard matrix is any complex N × N {displaystyle N imes N} matrix H {displaystyle H} satisfying two conditions: where † {displaystyle {dagger }} denotes the Hermitian transpose of H {displaystyle H} and I {displaystyle I} is the identity matrix. The concept is a generalization of the Hadamard matrix. Note that any complex Hadamard matrix H {displaystyle H} can be made into a unitary matrix by multiplying it by 1 N {displaystyle {frac {1}{sqrt {N}}}} ; conversely, any unitary matrix whose entries all have modulus 1 N {displaystyle {frac {1}{sqrt {N}}}} becomes a complex Hadamard upon multiplication by N {displaystyle {sqrt {N}}} . 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 {displaystyle N} (compare the real case, in which existence is not known for every N {displaystyle N} ). For instance the Fourier matrices (the complex conjugate of the DFT matrices without the normalizing factor),