matrix obtained by transposing
is formally defined by where the subscript
-th entry (matrix element), for
, and the overbar denotes a scalar complex conjugate.
denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate.
The conjugate transpose of a matrix
denotes the matrix with only complex conjugated entries and no transposition.
Suppose we want to calculate the conjugate transpose of the following matrix
are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix
The conjugate transpose of a matrix
with real entries reduces to the transpose of
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by
real matrices, obeying matrix addition and multiplication:
matrix of the linear transformation on the Argand diagram (viewed as the real vector space
matrix of complex numbers could be well represented by a
The conjugate transpose, therefore, arises very naturally as the result of simply transposing such a matrix—when viewed back again as an
matrix made up of complex numbers.
For an explanation of the notation used here, we begin by representing complex numbers
, we are led to the matrix representations of the unit numbers as
A general complex number
The complex conjugate operation (that sends
) is encoded as the matrix transpose.
as a linear transformation from Hilbert space
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.
is a linear map from a complex vector space
, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of
to be the complex conjugate of the transpose of
It maps the conjugate dual of