Conjugate Transpose

In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A 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

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, where a and b are reals, is .)

This definition can also be written as

where denotes the transpose and denotes the matrix with complex conjugated entries.

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

• or, commonly used in linear algebra
• (sometimes pronounced "A dagger"), universally used in quantum mechanics
• , although this symbol is more commonly used for the Moore-Penrose pseudoinverse

In some contexts, denotes the matrix with complex conjugated entries, and thus the conjugate transpose is denoted by or .