Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
"Adjoint matrix" redirects here. For the transpose of cofactor, see Adjugate matrix.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complexmatrix is an matrix obtained by transposing and applying complex conjugate on each entry (the complex conjugate of being , for real numbers and ). It is often denoted as or [1][2]
or
,[3] and very commonly in physics as .
For real matrices, the conjugate transpose is just the transpose, .
Definition
The conjugate transpose of an matrix is formally defined by
(Eq.1)
where the subscript denotes the -th entry, for and , and the overbar denotes a scalar complex conjugate.
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 can be denoted by any of these symbols:
The conjugate transpose "adjoint" matrix should not be confused with the adjugate, , which is also sometimes called adjoint.
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself.
Motivation
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by real matrices, obeying matrix addition and multiplication:
That is, denoting each complex number by the real matrix of the linear transformation on the Argand diagram (viewed as the real vector space ), affected by complex -multiplication on .
Thus, an 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 matrix made up of complex numbers.
Properties of the conjugate transpose
for any two matrices and of the same dimensions.
for any complex number and any matrix .
for any matrix and any matrix . Note that the order of the factors is reversed.[1]
for any matrix , i.e. Hermitian transposition is an involution.
If is a square matrix, then where denotes the determinant of .
If is a square matrix, then where denotes the trace of .
for any matrix , any vector in and any vector . Here, denotes the standard complex inner product on , and similarly for .
Generalizations
The last property given above shows that if one views as a linear transformation from Hilbert space to then the matrix corresponds to the adjoint operator of . 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 is a linear map from a complex vector space to another, , 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 to the conjugate dual of .