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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 (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

$$(\boldsymbol{A}^*)_{ij} = \overline{\boldsymbol{A}_{ji}}$$

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. (The complex conjugate of a + bi, where a and b are reals, is a - bi.)

This definition can also be written as

$$\boldsymbol{A}^* = (\overline{\boldsymbol{A}})^\mathrm{T} = \overline{\boldsymbol{A}^\mathrm{T}}$$

where$$\boldsymbol{A}^\mathrm{T} \,\!$$ denotes the transpose and $$\overline{\boldsymbol{A}} \,\!$$ 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:

$$\boldsymbol{A}^* \,\!$$ or $$\boldsymbol{A}^\mathrm{H} \,\!$$, commonly used in linear algebra
$$\boldsymbol{A}^\dagger \,\!$$ (sometimes pronounced as "A dagger"), universally used in quantum mechanics
$$\boldsymbol{A}^+ \,\!,$$ although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, $$\boldsymbol{A}^* \,\! denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by\( \boldsymbol{A}^{*\mathrm{T}} \,\!$$ or $$\boldsymbol{A}^{\mathrm{T}*} \,\!.$$

Example

If

$$\boldsymbol{A} = \begin{bmatrix} 1 & -2-i \\ 1+i & i \end{bmatrix}$$

then

$$\boldsymbol{A}^* = \begin{bmatrix} 1 & 1-i \\ -2+i & -i\end{bmatrix}$$

Basic remarks

A square matrix A with entries $$a_{ij}$$ is called

Hermitian or self-adjoint if A = A∗, i.e., $$a_{ij}=\overline{a_{ji}}$$ .
skew Hermitian or antihermitian if A = −A∗, i.e., $$a_{ij}=-\overline{a_{ji}}$$.
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.

Finding the conjugate transpose of a matrix A with real entries reduces to finding the transpose of A, 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 2×2 real matrices, obeying matrix addition and multiplication:

$$a + ib \equiv \left(\begin{matrix} a & -b \\ b & a \end{matrix}\right).$$

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 $$\mathbb{R}^2)$$affected by complex z-multiplication on $$\mathbb{C}$$.

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)∗ = r∗A∗ for any complex number r and any matrix A. Here, r∗ refers to the complex conjugate of r.
(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 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.
$$\langle A\boldsymbol{x}, \boldsymbol{y}\rangle = \langle \boldsymbol{x},A^* \boldsymbol{y} \rangle$$ for any m-by-n matrix A, any vector x in $$\mathbb{C}^n$$ and any vector y in $$\mathbb{C}^m$$. Here, $$\langle\cdot,\cdot\rangle$$ denotes the standard complex inner product on $$\mathbb{C}^m$$ and $$\mathbb{C}^n$$.

Generalizations

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.

Hermitian conjugate

Hazewinkel, Michiel, ed. (2001), "Adjoint matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Conjugate Transpose", MathWorld.
Conjugate transpose at PlanetMath.org.

Mathematics Encyclopedia