Conjugate transpose

"Adjoint matrix" redirects here. For the transpose of cofactor, see Adjugate matrix.

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

\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.

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

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 n-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 n-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 $A$ .
$\langle A\boldsymbol{x}, \boldsymbol{y}\rangle = \langle \boldsymbol{x},A^* \boldsymbol{y} \rangle$ 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 $C n$ to $C m$ , 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), "Adjoint matrix", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Conjugate Transpose", MathWorld.
Conjugate transpose at PlanetMath.org.

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