Conjugate transpose

In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix ${\displaystyle {\boldsymbol {A}}}$ with complex entries is the n-by-m matrix obtained from ${\displaystyle {\boldsymbol {A}}}$ by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of ${\displaystyle a+ib}$ being ${\displaystyle a-ib}$, for real numbers ${\displaystyle a}$ and ${\displaystyle b}$). It is often denoted as ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$ or ${\displaystyle {\boldsymbol {A}}^{*}}$.^[1][2]

For real matrices, the conjugate transpose is just the transpose, ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{\mathsf {T}}}$.

Definition[]

The conjugate transpose of an ${\displaystyle m\times n}$ matrix ${\displaystyle {\boldsymbol {A}}}$ is formally defined by

${\displaystyle \left({\boldsymbol {A}}^{\mathrm {H} }\right)_{ij}={\overline {{\boldsymbol {A}}_{ji}}}}$

(Eq.1)

where the subscript ${\displaystyle ij}$ denotes the ${\displaystyle (i,j)}$-th entry, for ${\displaystyle 1\leq i\leq n}$ and ${\displaystyle 1\leq j\leq m}$, and the overbar denotes a scalar complex conjugate.

This definition can also be written as^[2]

${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }=\left({\overline {\boldsymbol {A}}}\right)^{\mathsf {T}}={\overline {{\boldsymbol {A}}^{\mathsf {T}}}}}$

where ${\displaystyle {\boldsymbol {A}}^{\mathsf {T}}}$ 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 ${\displaystyle {\boldsymbol {A}}}$ can be denoted by any of these symbols:

• ${\displaystyle {\boldsymbol {A}}^{*}}$, commonly used in linear algebra^[2]
• ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$, commonly used in linear algebra
• ${\displaystyle {\boldsymbol {A}}^{\dagger }}$ (sometimes pronounced as A dagger), commonly 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 only complex conjugated entries and no transposition.

Example[]

Suppose we want to calculate the conjugate transpose of the following matrix ${\displaystyle {\boldsymbol {A}}}$.

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

We first transpose the matrix:

${\displaystyle {\boldsymbol {A}}^{\mathsf {T}}={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}}$

Then we conjugate every entry of the matrix:

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

Basic remarks[]

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

• Hermitian or self-adjoint if ${\displaystyle {\boldsymbol {A}}={\boldsymbol {A}}^{\mathrm {H} }}$; i.e., ${\displaystyle a_{ij}={\overline {a_{ji}}}}$.
• Skew Hermitian or antihermitian if ${\displaystyle {\boldsymbol {A}}=-{\boldsymbol {A}}^{\mathrm {H} }}$; i.e., ${\displaystyle a_{ij}=-{\overline {a_{ji}}}}$.
• Normal if ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }}$.
• Unitary if ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{-1}}$, equivalently ${\displaystyle {\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {I}}}$, equivalently ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {I}}}$.

Even if ${\displaystyle {\boldsymbol {A}}}$ is not square, the two matrices ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}}$ and ${\displaystyle {\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }}$ are both Hermitian and in fact positive semi-definite matrices.

The conjugate transpose "adjoint" matrix ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$ should not be confused with the adjugate, ${\displaystyle \operatorname {adj} ({\boldsymbol {A}})}$, which is also sometimes called adjoint.

The conjugate transpose of a matrix ${\displaystyle {\boldsymbol {A}}}$ with real entries reduces to the transpose of ${\displaystyle {\boldsymbol {A}}}$, 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:

${\displaystyle a+ib\equiv {\begin{bmatrix}a&-b\\b&a\end{bmatrix}}.}$

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 ${\displaystyle \mathbb {C} }$.

Thus, an m-by-n matrix of complex numbers could be well 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[]

• ${\displaystyle ({\boldsymbol {A}}+{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {A}}^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }}$ for any two matrices ${\displaystyle {\boldsymbol {A}}}$ and ${\displaystyle {\boldsymbol {B}}}$ of the same dimensions.
• ${\displaystyle (z{\boldsymbol {A}})^{\mathrm {H} }={\overline {z}}{\boldsymbol {A}}^{\mathrm {H} }}$ for any complex number ${\displaystyle z}$ and any m-by-n matrix ${\displaystyle {\boldsymbol {A}}}$.
• ${\displaystyle ({\boldsymbol {A}}{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {B}}^{\mathrm {H} }{\boldsymbol {A}}^{\mathrm {H} }}$ for any m-by-n matrix ${\displaystyle {\boldsymbol {A}}}$ and any n-by-p matrix ${\displaystyle {\boldsymbol {B}}}$. Note that the order of the factors is reversed.^[1]
• ${\displaystyle \left({\boldsymbol {A}}^{\mathrm {H} }\right)^{\mathrm {H} }={\boldsymbol {A}}}$ for any m-by-n matrix ${\displaystyle {\boldsymbol {A}}}$, i.e. Hermitian transposition is an involution.
• If ${\displaystyle {\boldsymbol {A}}}$ is a square matrix, then ${\displaystyle \det \left({\boldsymbol {A}}^{\mathrm {H} }\right)={\overline {\det \left({\boldsymbol {A}}\right)}}}$ where ${\displaystyle \operatorname {det} (A)}$ denotes the determinant of ${\displaystyle {\boldsymbol {A}}}$ .
• If ${\displaystyle {\boldsymbol {A}}}$ is a square matrix, then ${\displaystyle \operatorname {tr} \left({\boldsymbol {A}}^{\mathrm {H} }\right)={\overline {\operatorname {tr} ({\boldsymbol {A}})}}}$ where ${\displaystyle \operatorname {tr} (A)}$ denotes the trace of ${\displaystyle {\boldsymbol {A}}}$.
• ${\displaystyle {\boldsymbol {A}}}$ is invertible if and only if ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$ is invertible, and in that case ${\displaystyle \left({\boldsymbol {A}}^{\mathrm {H} }\right)^{-1}=\left({\boldsymbol {A}}^{-1}\right)^{\mathrm {H} }}$.
• The eigenvalues of ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$ are the complex conjugates of the eigenvalues of ${\displaystyle {\boldsymbol {A}}}$.
• ${\displaystyle \left\langle {\boldsymbol {A}}x,y\right\rangle _{m}=\left\langle x,{\boldsymbol {A}}^{\mathrm {H} }y\right\rangle _{n}}$ for any m-by-n matrix ${\displaystyle {\boldsymbol {A}}}$, any vector in ${\displaystyle x\in \mathbb {C} ^{n}}$ and any vector ${\displaystyle y\in \mathbb {C} ^{m}}$. Here, ${\displaystyle \langle \cdot ,\cdot \rangle _{m}}$ denotes the standard complex inner product on ${\displaystyle \mathbb {C} ^{m}}$, and similarly for ${\displaystyle \langle \cdot ,\cdot \rangle _{n}}$.

Generalizations[]

The last property given above shows that if one views ${\displaystyle {\boldsymbol {A}}}$ as a linear transformation from Hilbert space ${\displaystyle \mathbb {C} ^{n}}$ to ${\displaystyle \mathbb {C} ^{m},}$ then the matrix ${\displaystyle {\boldsymbol {A}}^{\mathrm {H} }}$ corresponds to the adjoint operator of ${\displaystyle {\boldsymbol {A}}}$. 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 ${\displaystyle A}$ is a linear map from a complex vector space ${\displaystyle V}$ to another, ${\displaystyle W}$, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of ${\displaystyle A}$ to be the complex conjugate of the transpose of ${\displaystyle A}$. It maps the conjugate dual of ${\displaystyle W}$ to the conjugate dual of ${\displaystyle V}$.

See also[]

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

References[]

External links[]

• "Adjoint matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994]