Conjugate transpose

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

\left({\boldsymbol {A}}^{*}\right)_{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.

This definition can also be written as

{\boldsymbol {A}}^{*}=\left({\overline {\boldsymbol {A}}}\right)^{\mathsf {T}}={\overline {{\boldsymbol {A}}^{\mathsf {T}}}}

where ${\boldsymbol {A}}^{\mathsf {T}}$ 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 ${\boldsymbol {A}}^{{*}{\mathsf {T}}}$ or ${\boldsymbol {A}}^{{\mathsf {T}}{*}}$ .

Example

If

{\boldsymbol {A}}={\begin{bmatrix}1&-2-i&5\\1+i&i&4-2i\end{bmatrix}}

then

{\boldsymbol {A}}^{\mathrm {H} }={\begin{bmatrix}1&1-i\\-2+i&-i\\5&4+2i\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.

The conjugate transpose "adjoint" matrix A^∗ should not be confused with the adjugate, adj(A), which is also sometimes called adjoint.

The conjugate transpose of a matrix A with real entries reduces to 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 {\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.

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)^∗ = 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 $\mathbb{C}^n$ 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 $\mathbb{C}^n$ to $\mathbb{C}^m$ , 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.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Adjoint matrix", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Weisstein, Eric W. "Conjugate Transpose". MathWorld.
"Conjugate transpose". PlanetMath.

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