Conjugate transpose

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an $$m \times n$$ complex matrix $$\mathbf{A}$$ is an $$n \times m$$ matrix obtained by transposing $$\mathbf{A}$$ and applying complex conjugation to each entry (the complex conjugate of $$a+ib$$ being $$a-ib$$, for real numbers $$a$$ and $$b$$). There are several notations, such as $$\mathbf{A}^\mathrm{H}$$ or $$\mathbf{A}^*$$, $$\mathbf{A}'$$, or (often in physics) $$\mathbf{A}^{\dagger}$$.

For real matrices, the conjugate transpose is just the transpose, $$\mathbf{A}^\mathrm{H} = \mathbf{A}^\operatorname{T}$$.

Definition
The conjugate transpose of an $$m \times n$$ matrix $$\mathbf{A}$$ is formally defined by

where the subscript $$ij$$ denotes the $$(i,j)$$-th entry, for $$1 \le i \le n$$ and $$1 \le j \le m$$, and the overbar denotes a scalar complex conjugate.

This definition can also be written as
 * $$\mathbf{A}^\mathrm{H} = \left(\overline{\mathbf{A}}\right)^\operatorname{T} = \overline{\mathbf{A}^\operatorname{T}}$$

where $$\mathbf{A}^\operatorname{T}$$ denotes the transpose and $$\overline{\mathbf{A}}$$ 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 $$\mathbf{A}$$ can be denoted by any of these symbols:
 * $$\mathbf{A}^*$$, commonly used in linear algebra
 * $$\mathbf{A}^\mathrm{H}$$, commonly used in linear algebra
 * $$\mathbf{A}^\dagger$$ (sometimes pronounced as A dagger), commonly used in quantum mechanics
 * $$\mathbf{A}^+$$, although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, $$\mathbf{A}^*$$ denotes the matrix with only complex conjugated entries and no transposition.

Example
Suppose we want to calculate the conjugate transpose of the following matrix $$\mathbf{A}$$.
 * $$\mathbf{A} = \begin{bmatrix} 1 & -2 - i & 5 \\ 1 + i & i & 4-2i \end{bmatrix}$$

We first transpose the matrix:
 * $$\mathbf{A}^\operatorname{T} = \begin{bmatrix} 1 & 1 + i \\ -2 - i & i \\ 5 & 4-2i\end{bmatrix}$$

Then we conjugate every entry of the matrix:
 * $$\mathbf{A}^\mathrm{H} = \begin{bmatrix} 1 & 1 - i \\ -2 + i & -i \\ 5 & 4+2i\end{bmatrix}$$

Basic remarks
A square matrix $$\mathbf{A}$$ with entries $$a_{ij}$$ is called
 * Hermitian or self-adjoint if $$\mathbf{A}=\mathbf{A}^\mathrm{H}$$; i.e., $$a_{ij} = \overline{a_{ji}}$$.
 * Skew Hermitian or antihermitian if $$\mathbf{A}=-\mathbf{A}^\mathrm{H}$$; i.e., $$a_{ij} = -\overline{a_{ji}}$$.
 * Normal if $$\mathbf{A}^\mathrm{H} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{H}$$.
 * Unitary if $$\mathbf{A}^\mathrm{H} = \mathbf{A}^{-1}$$, equivalently $$\mathbf{A}\mathbf{A}^\mathrm{H} = \boldsymbol{I}$$, equivalently $$\mathbf{A}^\mathrm{H}\mathbf{A} = \boldsymbol{I}$$.

Even if $$\mathbf{A}$$ is not square, the two matrices $$\mathbf{A}^\mathrm{H}\mathbf{A}$$ and $$\mathbf{A}\mathbf{A}^\mathrm{H}$$ are both Hermitian and in fact positive semi-definite matrices.

The conjugate transpose "adjoint" matrix $$\mathbf{A}^\mathrm{H}$$ should not be confused with the adjugate, $$\operatorname{adj}(\mathbf{A})$$, which is also sometimes called adjoint.

The conjugate transpose of a matrix $$\mathbf{A}$$ with real entries reduces to the transpose of $$\mathbf{A}$$, 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 $$2 \times 2$$ real matrices, obeying matrix addition and multiplication:
 * $$a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.$$

That is, denoting each complex number $$z$$ by the real $$2 \times 2$$ matrix of the linear transformation on the Argand diagram (viewed as the real vector space $$\mathbb{R}^2$$), affected by complex $$z$$-multiplication on $$\mathbb{C}$$.

Thus, an $$m \times n$$ matrix of complex numbers could be well represented by a $$2m \times 2n$$ 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 an $$n \times m$$ matrix made up of complex numbers.

For an explanation of the notation used here, we begin by representing complex numbers $$e^{i\theta}$$ as the rotation matrix, that is,

$$ e^{i\theta} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} = \cos \theta \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + \sin \theta \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. $$

Since $$ e^{i\theta} = \cos \theta + i \sin \theta$$ we are led to the matrix representations of the unit numbers as

$$ 1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}. $$ A general complex number $$z=x+iy$$ is then represented as

$$ z = \begin{pmatrix} x & -y \\ y & x \end{pmatrix}. $$ The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

Properties of the conjugate transpose

 * $$(\mathbf{A} + \boldsymbol{B})^\mathrm{H} = \mathbf{A}^\mathrm{H} + \boldsymbol{B}^\mathrm{H}$$ for any two matrices $$\mathbf{A}$$ and $$\boldsymbol{B}$$ of the same dimensions.
 * $$(z\mathbf{A})^\mathrm{H} = \overline{z} \mathbf{A}^\mathrm{H}$$ for any complex number $$z$$ and any $$m \times n$$ matrix $$\mathbf{A}$$.
 * $$(\mathbf{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \mathbf{A}^\mathrm{H}$$ for any $$m \times n$$ matrix $$\mathbf{A}$$ and any $$n \times p$$ matrix $$\boldsymbol{B}$$. Note that the order of the factors is reversed.
 * $$\left(\mathbf{A}^\mathrm{H}\right)^\mathrm{H} = \mathbf{A}$$ for any $$m \times n$$ matrix $$\mathbf{A}$$, i.e. Hermitian transposition is an involution.
 * If $$\mathbf{A}$$ is a square matrix, then $$\det\left(\mathbf{A}^\mathrm{H}\right) = \overline{\det\left(\mathbf{A}\right)}$$ where $$\operatorname{det}(A)$$ denotes the determinant of $$\mathbf{A}$$.
 * If $$\mathbf{A}$$ is a square matrix, then $$\operatorname{tr}\left(\mathbf{A}^\mathrm{H}\right) = \overline{\operatorname{tr}(\mathbf{A})}$$ where $$\operatorname{tr}(A)$$ denotes the trace of $$\mathbf{A}$$.
 * $$\mathbf{A}$$ is invertible if and only if $$\mathbf{A}^\mathrm{H}$$ is invertible, and in that case $$\left(\mathbf{A}^\mathrm{H}\right)^{-1} = \left(\mathbf{A}^{-1}\right)^{\mathrm{H}}$$.
 * The eigenvalues of $$\mathbf{A}^\mathrm{H}$$ are the complex conjugates of the eigenvalues of $$\mathbf{A}$$.
 * $$\left\langle \mathbf{A} x,y \right\rangle_m = \left\langle x, \mathbf{A}^\mathrm{H} y\right\rangle_n $$ for any $$m \times n$$ matrix $$\mathbf{A}$$, any vector in $$x \in \mathbb{C}^n $$ and any vector $$y \in \mathbb{C}^m $$. Here, $$\langle\cdot,\cdot\rangle_m$$ denotes the standard complex inner product on $$ \mathbb{C}^m $$, and similarly for $$\langle\cdot,\cdot\rangle_n$$.

Generalizations
The last property given above shows that if one views $$\mathbf{A}$$ as a linear transformation from Hilbert space $$ \mathbb{C}^n $$ to $$ \mathbb{C}^m ,$$ then the matrix $$\mathbf{A}^\mathrm{H}$$ corresponds to the adjoint operator of $$\mathbf A$$. 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 $$V$$.