In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix 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 .
For real matrices, the conjugate transpose is just the transpose, .
The conjugate transpose of an matrix is formally defined by
where the subscript denotes the -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 can be denoted by any of these symbols:
- , commonly used in linear algebra
- , commonly used in linear algebra
- (sometimes pronounced as A dagger), commonly used in quantum mechanics
- , although this symbol is more commonly used for the Moore–Penrose pseudoinverse
In some contexts, denotes the matrix with only complex conjugated entries and no transposition.
Suppose we want to calculate the conjugate transpose of the following matrix .
We first transpose the matrix:
Then we conjugate every entry of the matrix:
A square matrix with entries is called
- Hermitian or self-adjoint if ; i.e., .
- Skew Hermitian or antihermitian if ; i.e., .
- Normal if .
- Unitary if , equivalently , equivalently .
Even if 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 with real entries reduces to the transpose of , 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 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.
- 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 .
- is invertible if and only if is invertible, and in that case .
- The eigenvalues of are the complex conjugates of the eigenvalues of .
- for any matrix , any vector in and any vector . Here, denotes the standard complex inner product on , and similarly for .
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 .