# Conjugate transpose

(Redirected from Conjugate matrix)
"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

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

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 ${\displaystyle a+bi}$, where a and b are reals, is ${\displaystyle a-bi}$.)

This definition can also be written as

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

where ${\displaystyle {\boldsymbol {A}}^{\mathrm {T} }}$ denotes the transpose and ${\displaystyle {\overline {\boldsymbol {A}}}}$ 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:

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

In some contexts, ${\displaystyle {\boldsymbol {A}}^{*}}$ denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by ${\displaystyle {\boldsymbol {A}}^{{*}\mathrm {T} }}$ or ${\displaystyle {\boldsymbol {A}}^{\mathrm {T} {*}}}$.

## Example

If

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

then

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

## Basic remarks

A square matrix A with entries ${\displaystyle a_{ij}}$ is called

• Hermitian or self-adjoint if A = A, i.e., ${\displaystyle a_{ij}={\overline {a_{ji}}}}$ .
• skew Hermitian or antihermitian if A = −A, i.e., ${\displaystyle a_{ij}=-{\overline {a_{ji}}}}$ .
• normal if AA = AA.
• unitary if A = A−1.

Even if A is not square, the two matrices AA and AA are both Hermitian and in fact positive semi-definite matrices.

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:

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

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 ${\displaystyle \mathbb {R} ^{2}}$) affected by complex z-multiplication on ${\displaystyle \mathbb {C} }$.

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) = BA 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.
• ${\displaystyle \langle A{\boldsymbol {x}},{\boldsymbol {y}}\rangle =\langle {\boldsymbol {x}},A^{*}{\boldsymbol {y}}\rangle }$ for any m-by-n matrix A, any vector x in ${\displaystyle \mathbb {C} ^{n}}$ and any vector y in ${\displaystyle \mathbb {C} ^{m}}$. Here, ${\displaystyle \langle \cdot ,\cdot \rangle }$ denotes the standard complex inner product on ${\displaystyle \mathbb {C} ^{m}}$ and ${\displaystyle \mathbb {C} ^{n}}$.

## Generalizations

The last property given above shows that if one views A as a linear transformation from Euclidean Hilbert space Cn to Cm, 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.