What is "conjugate transpose"

n. (context mathematics English) transpose conjugate

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix with complex entries is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

$$(\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}}$$
where the subscripts denote the i,j-th entry, for 1 ≤ i ≤ n and 1 ≤ j ≤ m, and the overbar denotes a scalar complex conjugate. (The complex conjugate of a + bi, where a and b are reals, is a − bi.)

This definition can also be written as

$$\boldsymbol{A}^* = (\overline{\boldsymbol{A}})^\mathrm{T} = \overline{\boldsymbol{A}^\mathrm{T}}$$
where $\boldsymbol{A}^\mathrm{T} \,\!$ denotes the transpose and $\overline{\boldsymbol{A}} \,\!$ denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian conjugate, bedaggered matrix, adjoint matrix or transjugate. The conjugate transpose of a matrix can be denoted by any of these symbols:

• $\boldsymbol{A}^* \,\!$ or $\boldsymbol{A}^\mathrm{H} \,\!$, commonly used in linear algebra
• $\boldsymbol{A}^\dagger \,\!$ (sometimes pronounced as " dagger"), universally used in quantum mechanics
• $\boldsymbol{A}^+ \,\!$, although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, $\boldsymbol{A}^* \,\!$ denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by $\boldsymbol{A}^{*\mathrm{T}} \,\!$ or $\boldsymbol{A}^{\mathrm{T}*} \,\!$.