Conjugate transpose

Short description: Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry

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

For real matrices, the conjugate transpose is just the transpose, [math]\displaystyle{ \mathbf{A}^\mathrm{H} = \mathbf{A}^\operatorname{T} }[/math].

Definition

The conjugate transpose of an [math]\displaystyle{ m \times n }[/math] matrix [math]\displaystyle{ \mathbf{A} }[/math] is formally defined by

[math]\displaystyle{ \left(\mathbf{A}^\mathrm{H}\right)_{ij} = \overline{\mathbf{A}_{ji}} }[/math]

(Eq.1)

where the subscript [math]\displaystyle{ ij }[/math] denotes the [math]\displaystyle{ (i,j) }[/math]-th entry, for [math]\displaystyle{ 1 \le i \le n }[/math] and [math]\displaystyle{ 1 \le j \le m }[/math], and the overbar denotes a scalar complex conjugate.

This definition can also be written as

[math]\displaystyle{ \mathbf{A}^\mathrm{H} = \left(\overline{\mathbf{A}}\right)^\operatorname{T} = \overline{\mathbf{A}^\operatorname{T}} }[/math]

where [math]\displaystyle{ \mathbf{A}^\operatorname{T} }[/math] denotes the transpose and [math]\displaystyle{ \overline{\mathbf{A}} }[/math] 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 [math]\displaystyle{ \mathbf{A} }[/math] can be denoted by any of these symbols:

• [math]\displaystyle{ \mathbf{A}^* }[/math], commonly used in linear algebra
• [math]\displaystyle{ \mathbf{A}^\mathrm{H} }[/math], commonly used in linear algebra
• [math]\displaystyle{ \mathbf{A}^\dagger }[/math] (sometimes pronounced as A dagger), commonly used in quantum mechanics
• [math]\displaystyle{ \mathbf{A}^+ }[/math], although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, [math]\displaystyle{ \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]\displaystyle{ \mathbf{A} }[/math].

[math]\displaystyle{ \mathbf{A} = \begin{bmatrix} 1 & -2 - i & 5 \\ 1 + i & i & 4-2i \end{bmatrix} }[/math]

We first transpose the matrix:

[math]\displaystyle{ \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]\displaystyle{ \mathbf{A}^\mathrm{H} = \begin{bmatrix} 1 & 1 - i \\ -2 + i & -i \\ 5 & 4+2i\end{bmatrix} }[/math]

Basic remarks

A square matrix [math]\displaystyle{ \mathbf{A} }[/math] with entries [math]\displaystyle{ a_{ij} }[/math] is called

• Hermitian or self-adjoint if [math]\displaystyle{ \mathbf{A}=\mathbf{A}^\mathrm{H} }[/math]; i.e., [math]\displaystyle{ a_{ij} = \overline{a_{ji}} }[/math].
• Skew Hermitian or antihermitian if [math]\displaystyle{ \mathbf{A}=-\mathbf{A}^\mathrm{H} }[/math]; i.e., [math]\displaystyle{ a_{ij} = -\overline{a_{ji}} }[/math].
• Normal if [math]\displaystyle{ \mathbf{A}^\mathrm{H} \mathbf{A} = \mathbf{A} \mathbf{A}^\mathrm{H} }[/math].
• Unitary if [math]\displaystyle{ \mathbf{A}^\mathrm{H} = \mathbf{A}^{-1} }[/math], equivalently [math]\displaystyle{ \mathbf{A}\mathbf{A}^\mathrm{H} = \boldsymbol{I} }[/math], equivalently [math]\displaystyle{ \mathbf{A}^\mathrm{H}\mathbf{A} = \boldsymbol{I} }[/math].

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

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

The conjugate transpose of a matrix [math]\displaystyle{ \mathbf{A} }[/math] with real entries reduces to the transpose of [math]\displaystyle{ \mathbf{A} }[/math], 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 [math]\displaystyle{ 2 \times 2 }[/math] real matrices, obeying matrix addition and multiplication:

[math]\displaystyle{ a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}. }[/math]

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

Thus, an [math]\displaystyle{ m \times n }[/math] matrix of complex numbers could be well represented by a [math]\displaystyle{ 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]\displaystyle{ n \times m }[/math] matrix made up of complex numbers.

Properties of the conjugate transpose

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

Generalizations

The last property given above shows that if one views [math]\displaystyle{ \mathbf{A} }[/math] as a linear transformation from Hilbert space [math]\displaystyle{ \mathbb{C}^n }[/math] to [math]\displaystyle{ \mathbb{C}^m , }[/math] then the matrix [math]\displaystyle{ \mathbf{A}^\mathrm{H} }[/math] corresponds to the adjoint operator of [math]\displaystyle{ \mathbf A }[/math]. 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 [math]\displaystyle{ A }[/math] is a linear map from a complex vector space [math]\displaystyle{ V }[/math] to another, [math]\displaystyle{ W }[/math], then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of [math]\displaystyle{ A }[/math] to be the complex conjugate of the transpose of [math]\displaystyle{ A }[/math]. It maps the conjugate dual of [math]\displaystyle{ W }[/math] to the conjugate dual of [math]\displaystyle{ V }[/math].

See also

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

External links

• Hazewinkel, Michiel, ed. (2001), "Adjoint matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/a010850