In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf{A} is an n \times m matrix obtained by transposing \mathbf{A} and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). There are several notations, such as or, \mathbf{A}', or (often in physics). For real matrices, the conjugate transpose is just the transpose,.

Definition

The conjugate transpose of an m \times n matrix \mathbf{A} is formally defined by where the subscript ij denotes the (i,j)-th entry, for and, and the overbar denotes a scalar complex conjugate. 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, adjoint matrix or transjugate. The conjugate transpose of a matrix \mathbf{A} can be denoted by any of these symbols: In some contexts, denotes the matrix with only complex conjugated entries and no transposition.

Example

Suppose we want to calculate the conjugate transpose of the following matrix \mathbf{A}. We first transpose the matrix: Then we conjugate every entry of the matrix:

Basic remarks

A square matrix \mathbf{A} with entries a_{ij} is called Even if \mathbf{A} is not square, the two matrices and are both Hermitian and in fact positive semi-definite matrices. The conjugate transpose "adjoint" matrix should not be confused with the adjugate,, which is also sometimes called adjoint. The conjugate transpose of a matrix \mathbf{A} with real entries reduces to the transpose of \mathbf{A}, as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 \times 2 real matrices, obeying matrix addition and multiplication: That is, denoting each complex number z by the real 2 \times 2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex z-multiplication on \mathbb{C}. Thus, an m \times n matrix of complex numbers could be well represented by a matrix of real numbers. The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an n \times m matrix made up of complex numbers. For an explanation of the notation used here, we begin by representing complex numbers e^{i\theta} as the rotation matrix, that is, Since we are led to the matrix representations of the unit numbers as A general complex number z=x+iy is then represented as The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

Properties

Generalizations

The last property given above shows that if one views \mathbf{A} as a linear transformation from Hilbert space to then the matrix corresponds to the adjoint operator of \mathbf A. The concept of adjoint operators between Hilbert spaces can thus be seen as a generalization of the conjugate transpose of matrices with respect to an orthonormal basis. Another generalization is available: suppose A is a linear map from a complex vector space V to another, W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A. It maps the conjugate dual of W to the conjugate dual of V.