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

If

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

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 matrix A. Here, r^∗ refers to the complex conjugate of r.
• (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 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 Cⁿ 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.

See also

• Hermitian conjugate
• Adjugate matrix

External links

• Weisstein, Eric W., "Conjugate Transpose", MathWorld.
• Conjugate transpose at PlanetMath.org.