Conjugate transpose

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an ${\displaystyle m\times n}${\displaystyle m\times n} complex matrix ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} } is an ${\displaystyle n\times m}${\displaystyle n\times m} matrix obtained by transposing ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} } and applying complex conjugation to each entry (the complex conjugate of ${\displaystyle a+ib}${\displaystyle a+ib} being ${\displaystyle a-ib}${\displaystyle a-ib}, for real numbers ${\displaystyle a}${\displaystyle a} and ${\displaystyle b}${\displaystyle b}). There are several notations, such as ${\displaystyle \mathbf {A} ^{\mathrm {H} }}${\displaystyle \mathbf {A} ^{\mathrm {H} }} or ${\displaystyle \mathbf {A} ^{*}}${\displaystyle \mathbf {A} ^{*}},^[1] ${\displaystyle \mathbf {A} '}${\displaystyle \mathbf {A} '},^[2] or (often in physics) ${\displaystyle \mathbf {A} ^{\dagger }}${\displaystyle \mathbf {A} ^{\dagger }}.

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

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

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

This definition can also be written as

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

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

Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} } can be denoted by any of these symbols:

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

In some contexts, ${\displaystyle \mathbf {A} ^{*}}${\displaystyle \mathbf {A} ^{*}} denotes the matrix with only complex conjugated entries and no transposition.

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

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

We first transpose the matrix:

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

Then we conjugate every entry of the matrix:

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

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

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

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

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

The conjugate transpose of a matrix ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} } with real entries reduces to the transpose of ${\displaystyle \mathbf {A} }${\displaystyle \mathbf {A} }, as the conjugate of a real number is the number itself.

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by ${\displaystyle 2\times 2}${\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication:^[3]

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

That is, denoting each complex number ${\displaystyle z}${\displaystyle z} by the real ${\displaystyle 2\times 2}${\displaystyle 2\times 2} matrix of the linear transformation on the Argand diagram (viewed as the real vector space ${\displaystyle \mathbb {R} ^{2}}${\displaystyle \mathbb {R} ^{2}}), affected by complex ${\displaystyle z}${\displaystyle z}-multiplication on ${\displaystyle \mathbb {C} }${\displaystyle \mathbb {C} }.

Thus, an ${\displaystyle m\times n}${\displaystyle m\times n} matrix of complex numbers could be well represented by a ${\displaystyle 2m\times 2n}${\displaystyle 2m\times 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 an ${\displaystyle n\times m}${\displaystyle n\times m} matrix made up of complex numbers.

For an explanation of the notation used here, we begin by representing complex numbers ${\displaystyle e^{i\theta }}${\displaystyle e^{i\theta }} as the rotation matrix, that is,

${\displaystyle e^{i\theta }={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}=\cos \theta {\begin{pmatrix}1&0\0円&1\end{pmatrix}}+\sin \theta {\begin{pmatrix}0&-1\1円&0\end{pmatrix}}.}${\displaystyle e^{i\theta }={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}=\cos \theta {\begin{pmatrix}1&0\0円&1\end{pmatrix}}+\sin \theta {\begin{pmatrix}0&-1\1円&0\end{pmatrix}}.}

Since ${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }${\displaystyle e^{i\theta }=\cos \theta +i\sin \theta }, we are led to the matrix representations of the unit numbers as

${\displaystyle 1={\begin{pmatrix}1&0\0円&1\end{pmatrix}},\quad i={\begin{pmatrix}0&-1\1円&0\end{pmatrix}}.}${\displaystyle 1={\begin{pmatrix}1&0\0円&1\end{pmatrix}},\quad i={\begin{pmatrix}0&-1\1円&0\end{pmatrix}}.}

A general complex number ${\displaystyle z=x+iy}${\displaystyle z=x+iy} is then represented as ${\displaystyle z={\begin{pmatrix}x&-y\\y&x\end{pmatrix}}.}${\displaystyle z={\begin{pmatrix}x&-y\\y&x\end{pmatrix}}.}

The complex conjugate operation, where i→−i, is seen to be just the matrix transpose.

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

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

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

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

• Linear algebra
• Matrices

• Articles with short description
• Short description matches Wikidata