In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an <math>m \times n</math> complex matrix <math>\mathbf{A}</math> is an <math>n \times m</math> matrix obtained by transposing <math>\mathbf{A}</math> and applying complex conjugation to each entry (the complex conjugate of <math>a+ib</math> being <math>a-ib</math>, for real numbers <math>a</math> and <math>b</math>). There are several notations, such as <math>\mathbf{A}^\mathrm{H}</math> or <math>\mathbf{A}^*</math>, <math>\mathbf{A}'</math>, or (often in physics) <math>\mathbf{A}^{\dagger}</math>.
For real matrices, the conjugate transpose is just the transpose, <math>\mathbf{A}^\mathrm{H} = \mathbf{A}^\operatorname{T}</math>.
Definition
The conjugate transpose of an <math>m \times n</math> matrix <math>\mathbf{A}</math> is formally defined by
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where the subscript <math>ij</math> denotes the <math>(i,j)</math>-th entry (matrix element), for <math>1 \le i \le n</math> and <math>1 \le j \le m</math>, and the overbar denotes a scalar complex conjugate.
This definition can also be written as
:<math>\mathbf{A}^\mathrm{H} = \left(\overline{\mathbf{A\right)^\operatorname{T} = \overline{\mathbf{A}^\operatorname{T</math>
where <math>\mathbf{A}^\operatorname{T}</math> denotes the transpose and <math>\overline{\mathbf{A</math> denotes the matrix with complex conjugated entries.
Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix <math>\mathbf{A}</math> can be denoted by any of these symbols:
- <math>\mathbf{A}^*</math>, commonly used in linear algebra
- <math>\mathbf{A}^\mathrm{H}</math>, commonly used in linear algebra
- <math>\mathbf{A}^\dagger</math> (sometimes pronounced as A dagger), commonly used in quantum mechanics
- <math>\mathbf{A}^+</math>, although this symbol is more commonly used for the Moore–Penrose pseudoinverse
In some contexts, <math>\mathbf{A}^*</math> denotes the matrix with only complex conjugated entries and no transposition.
Example
Suppose we want to calculate the conjugate transpose of the following matrix <math>\mathbf{A}</math>.
:<math>\mathbf{A} = \begin{bmatrix} 1 & -2 - i & 5 \\ 1 + i & i & 4-2i \end{bmatrix}</math>
We first transpose the matrix:
:<math>\mathbf{A}^\operatorname{T} = \begin{bmatrix} 1 & 1 + i \\ -2 - i & i \\ 5 & 4-2i\end{bmatrix}</math>
Then we conjugate every entry of the matrix:
:<math>\mathbf{A}^\mathrm{H} = \begin{bmatrix} 1 & 1 - i \\ -2 + i & -i \\ 5 & 4+2i\end{bmatrix}</math>
Basic remarks
A square matrix <math>\mathbf{A}</math> with entries <math>a_{ij}</math> is called
- Hermitian or self-adjoint if <math>\mathbf{A}=\mathbf{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = \overline{a_{ji</math>.
- Skew Hermitian or antihermitian if <math>\mathbf{A}=-\mathbf{A}^\mathrm{H}</math>; i.e., <math>a_{ij} = -\overline{a_{ji</math>.
- Normal if <math>\mathbf{A}^\mathrm{H} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{H}</math>.
- Unitary if <math>\mathbf{A}^\mathrm{H} = \mathbf{A}^{-1}</math>, equivalently <math>\mathbf{A}\mathbf{A}^\mathrm{H} = \boldsymbol{I}</math>, equivalently <math>\mathbf{A}^\mathrm{H}\mathbf{A} = \boldsymbol{I}</math>.
Even if <math>\mathbf{A}</math> is not square, the two matrices <math>\mathbf{A}^\mathrm{H}\mathbf{A}</math> and <math>\mathbf{A}\mathbf{A}^\mathrm{H}</math> are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix <math>\mathbf{A}^\mathrm{H}</math> should not be confused with the adjugate, <math>\operatorname{adj}(\mathbf{A})</math>, which is also sometimes called adjoint.
The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by <math>2 \times 2</math> real matrices, obeying matrix addition and multiplication:
<math display="block">a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.</math>
That is, denoting each complex number <math>z</math> by the real <math>2 \times 2</math> matrix of the linear transformation on the Argand diagram (viewed as the real vector space <math>\mathbb{R}^2</math>), affected by complex <math>z</math>-multiplication on <math>\mathbb{C}</math>.
Thus, an <math>m \times n</math> matrix of complex numbers could be well represented by a <math>2m \times 2n</math> 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 <math>n \times m</math> matrix made up of complex numbers.
For an explanation of the notation used here, we begin by representing complex numbers <math>e^{i\theta}</math> as the rotation matrix, that is,
<math display="block">
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}.
</math>
Since <math>e^{i\theta} = \cos \theta + i \sin \theta</math>, we are led to the matrix representations of the unit numbers as
<math display="block">
1 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad i = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.
</math>
A general complex number <math>z=x+iy</math> is then represented as <math>
z = \begin{pmatrix} x & -y \\ y & x \end{pmatrix}.
</math> The complex conjugate operation (that sends <math>a + bi</math> to <math>a - bi</math> for real <math>a, b</math>) is encoded as the matrix transpose.
Properties
- <math>(\mathbf{A} + \boldsymbol{B})^\mathrm{H} = \mathbf{A}^\mathrm{H} + \boldsymbol{B}^\mathrm{H}</math> for any two matrices <math>\mathbf{A}</math> and <math>\boldsymbol{B}</math> of the same dimensions.
- <math>(z\mathbf{A})^\mathrm{H} = \overline{z} \mathbf{A}^\mathrm{H}</math> for any complex number <math>z</math> and any <math>m \times n</math> matrix <math>\mathbf{A}</math>.
- <math>(\mathbf{A}\boldsymbol{B})^\mathrm{H} = \boldsymbol{B}^\mathrm{H} \mathbf{A}^\mathrm{H}</math> for any <math>m \times n</math> matrix <math>\mathbf{A}</math> and any <math>n \times p</math> matrix <math>\boldsymbol{B}</math>. Note that the order of the factors is reversed.