# Conjugate transpose

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Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix 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

$(\mathbf{A}^*)_{ij} = \overline{\mathbf{A}_{ji}}$

where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate. (The complex conjugate of a + bi, where a and b are reals, is abi.)

This definition can also be written as

$\mathbf{A}^* = (\overline{\mathbf{A}})^\mathrm{T} = \overline{\mathbf{A}^\mathrm{T}}$

where $\mathbf{A}^\mathrm{T} \,\!$ denotes the transpose and $\overline{\mathbf{A}} \,\!$ denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian conjugate, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

• $\mathbf{A}^* \,\!$ or $\mathbf{A}^\mathrm{H} \,\!$, commonly used in linear algebra
• $\mathbf{A}^\dagger \,\!$ (sometimes pronounced "A dagger"), universally used in quantum mechanics
• $\mathbf{A}^+ \,\!$, although this symbol is more commonly used for the Moore-Penrose pseudoinverse

In some contexts, $\mathbf{A}^* \,\!$ denotes the matrix with complex conjugated entries, and thus the conjugate transpose is denoted by $\mathbf{A}^{*T} \,\!$ or $\mathbf{A}^{T*} \,\!$.

## Example

If

$\mathbf{A} = \begin{bmatrix} 3 + i & 5 \\ 2-2i & i \end{bmatrix}$

then

$\mathbf{A}^* = \begin{bmatrix} 3-i & 2+2i \\ 5 & -i \end{bmatrix}.$

## Basic remarks

A square matrix A with entries aij 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.

## 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:

$a + ib \equiv \Big(\begin{matrix} a & -b \\ b & a \end{matrix}\Big).$

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.
• (r A)* = 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 we have (A*)−1 = (A−1)*.
• The eigenvalues of A* are the complex conjugates of the eigenvalues of A.
• $\langle \mathbf{Ax}, \mathbf{y}\rangle = \langle \mathbf{x},\mathbf{A}^* \mathbf{y} \rangle$ 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 the 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.

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.

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