= Hermitian Transpose = The '''Hermitian transpose''' is [[LinearAlgebra/Transposition|transposition]] then [[LinearAlgebra/MatrixConjugation|conjugation]]. <> ---- == Description == The Hermitian transpose of a matrix '''''A''''' is calculated by [[LinearAlgebra/Transposition|transposing]] it and taking the complex [[LinearAlgebra/MatrixConjugation|conjugate]]. There isn't a standard notation: use one of '''''A'''^H^'' or '''''A'''^†^'' ('dagger') or '''''A'''^*^''. If '''''A'''^H^ = '''A'''^-1^'', or if '''''AA'''^H^ = '''A'''^H^'''A''' = '''I''''', then '''''A''''' is '''unitary'''. If '''''A'''^H^ = '''A''''', then '''''A''''' is '''Hermitian'''. This looks like a symmetric matrix where the complex components mirrored across the diagonal have flipped signs. Only a square matrix can be Hermitian. === Properties === The double transpose of a matrix is the original matrix: '''''A'''^*^^*^ = '''A'''''. ''('''A''' + '''B''')^*^ = '''A'''^*^ + '''B'''^*^''. ''('''AB''')^*^ = '''B'''^*^'''A'''^*^'' ''(c'''A''')^*^ = c̅('''A''')^*^'' The product of two unitary matrices is also unitary. ---- CategoryRicottone