In mathematics, the conjugate transpose or Hermitian transpose 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. (The complex conjugate of a + bi, where a and b are reals, is a − bi.) 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.
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, bedaggered matrix, adjoint matrix 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 as "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 the conjugate transpose is then denoted by or .
A square matrix A with entries is called
- Hermitian or self-adjoint if A = A∗, i.e. .
- skew Hermitian or antihermitian if A = −A∗, i.e. .
- normal if A∗A = AA∗.
- unitary if A∗ = A−1.
Even if A is not square, the two matrices A∗A and AA∗ are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix A∗ should not be confused with the adjugate adj(A), which is also sometimes called "adjoint".
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2×2 real matrices, obeying matrix addition and multiplication:
That is, denoting each complex number z by the real 2×2 matrix of the linear transformation on the Argand diagram (viewed as the real vector space ) affected by complex z-multiplication on .
An m-by-n matrix of complex numbers could therefore equally well be represented by a 2m-by-2n 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 n-by-m matrix made up of complex numbers.
Properties of the conjugate transpose
- (A + B)∗ = A∗ + B∗ for any two matrices A and B of the same dimensions.
- (rA)∗ = rA∗ for any complex number r and any m-by-n matrix A.
- (AB)∗ = B∗A∗ for any m-by-n matrix A and any n-by-p matrix B. Note that the order of the factors is reversed.
- (A∗)∗ = A for any m-by-n matrix A.
- If A is a square matrix, then det(A∗) = (det A)∗ and tr(A∗) = (tr A)∗.
- A is invertible if and only if A∗ is invertible, and in that case (A∗)−1 = (A−1)∗.
- The eigenvalues of A∗ are the complex conjugates of the eigenvalues of A.
- ⟨Ax, y⟩ = ⟨x, A∗y⟩ for any m-by-n matrix A, any vector x in and any vector y in . Here, denotes the standard complex inner product on and .
The last property given above shows that if one views A as a linear transformation from Euclidean Hilbert space Cn to Cm, then the matrix A∗ corresponds to the adjoint operator of 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.