Conjugate Transpose

The conjugate transpose of an matrix is the matrix defined by

(1)

where denotes the transpose of the matrix and denotes the conjugate matrix. In all common spaces (i.e., separable Hilbert spaces), the conjugate and transpose operations commute, so

(2)

The symbol (where the "H" stands for "Hermitian") gives official recognition to the fact that for complex matrices, it is almost always the case that the combined operation of taking the transpose and complex conjugate arises in physical or computation contexts and virtually never the transpose in isolation (Strang 1988, pp. 220-221).

The conjugate transpose of a matrix is implemented in the Wolfram Language as ConjugateTranspose[A].

The conjugate transpose is also known as the adjoint matrix, adjugate matrix, Hermitian adjoint, or Hermitian transpose (Strang 1988, p. 221). Unfortunately, several different notations are in use as summarized in the following table. While the notation is universally used in quantum field theory, is commonly used in linear algebra. Note that because is sometimes used to denote the complex conjugate, special care must be taken not to confuse notations from different sources.

notation	references
	This work; Golub and van Loan (1996, p. 14), Strang (1988, p. 220)
	Courant and Hilbert (1989, p. 9), Lancaster and Tismenetsky (1984), Meyer (2000)
	Arfken (1985, p. 210), Weinberg (1995, p. xxv)

If a matrix is equal to its own conjugate transpose, it is said to be self-adjoint and is called a Hermitian.

The conjugate transpose of a matrix product is given by

(3)

Using the identity for the product of transpose gives

			(4)
			(5)
			(6)
			(7)
			(8)

where Einstein summation has been used here to sum over repeated indices, it follows that

(9)