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Conjugate transpose
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<h2 id="definition">Definition</h2> <p>The conjugate transpose of an m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } is formally defined by </p> <table><tbody><tr><td> ( A H ) i j = A j i ¯ {\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)_{ij}={\overline {\mathbf {A} _{ji}}}} </td> <td></td> <td>Eq.1</td></tr></tbody></table> <p>where the subscript i j {\displaystyle ij} denotes the ( i , j ) {\displaystyle (i,j)} -th entry (matrix element), for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} and 1 ≤ j ≤ m {\displaystyle 1\leq j\leq m} , and the overbar denotes a scalar complex conjugate. </p><p>This definition can also be written as </p> A H = ( A ¯ ) T = A T ¯ {\displaystyle \mathbf {A} ^{\mathrm {H} }=\left({\overline {\mathbf {A} }}\right)^{\operatorname {T} }={\overline {\mathbf {A} ^{\operatorname {T} }}}} <p>where A T {\displaystyle \mathbf {A} ^{\operatorname {T} }} denotes the transpose and A ¯ {\displaystyle {\overline {\mathbf {A} }}} denotes the matrix with complex conjugated entries. </p><p>Other names for the conjugate transpose of a matrix are Hermitian transpose, Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix A {\displaystyle \mathbf {A} } can be denoted by any of these symbols: </p> <ul><li> A ∗ {\displaystyle \mathbf {A} ^{*}} , commonly used in <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a></li> <li> A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} , commonly used in linear algebra</li> <li> A † {\displaystyle \mathbf {A} ^{\dagger }} (sometimes pronounced as <i>A <a href="/facts/Dagger_(typography)/lDNNgbgr">dagger</a></i>), commonly used in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a></li> <li> A + {\displaystyle \mathbf {A} ^{+}} , although this symbol is more commonly used for the <a href="/facts/Moore%25E2%2580%2593Penrose_pseudoinverse/CU73tntq">Moore–Penrose pseudoinverse</a></li></ul> <p>In some contexts, A ∗ {\displaystyle \mathbf {A} ^{*}} denotes the matrix with only complex conjugated entries and no transposition. </p> <h2 id="example">Example</h2> <p>Suppose we want to calculate the conjugate transpose of the following matrix A {\displaystyle \mathbf {A} } . </p> A = [ 1 − 2 − i 5 1 + i i 4 − 2 i ] {\displaystyle \mathbf {A} ={\begin{bmatrix}1&-2-i&5\\1+i&i&4-2i\end{bmatrix}}} <p>We first transpose the matrix: </p> A T = [ 1 1 + i − 2 − i i 5 4 − 2 i ] {\displaystyle \mathbf {A} ^{\operatorname {T} }={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}} <p>Then we conjugate every entry of the matrix: </p> A H = [ 1 1 − i − 2 + i − i 5 4 + 2 i ] {\displaystyle \mathbf {A} ^{\mathrm {H} }={\begin{bmatrix}1&1-i\\-2+i&-i\\5&4+2i\end{bmatrix}}} <h2 id="basic-remarks">Basic remarks</h2> <p>A square matrix A {\displaystyle \mathbf {A} } with entries a i j {\displaystyle a_{ij}} is called </p> <ul><li><a href="/facts/Hermitian_matrix/qArPblYo">Hermitian</a> or <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint</a> if A = A H {\displaystyle \mathbf {A} =\mathbf {A} ^{\mathrm {H} }} ; i.e., a i j = a j i ¯ {\displaystyle a_{ij}={\overline {a_{ji}}}} .</li> <li><a href="/facts/Skew-Hermitian_matrix/wxjjnvCC">Skew Hermitian</a> or antihermitian if A = − A H {\displaystyle \mathbf {A} =-\mathbf {A} ^{\mathrm {H} }} ; i.e., a i j = − a j i ¯ {\displaystyle a_{ij}=-{\overline {a_{ji}}}} .</li> <li><a href="/facts/Normal_matrix/WVKxrEK9">Normal</a> if A H A = A A H {\displaystyle \mathbf {A} ^{\mathrm {H} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {H} }} .</li> <li><a href="/facts/Unitary_matrix/8fsxv3KD">Unitary</a> if A H = A − 1 {\displaystyle \mathbf {A} ^{\mathrm {H} }=\mathbf {A} ^{-1}} , equivalently A A H = I {\displaystyle \mathbf {A} \mathbf {A} ^{\mathrm {H} }={\boldsymbol {I}}} , equivalently A H A = I {\displaystyle \mathbf {A} ^{\mathrm {H} }\mathbf {A} ={\boldsymbol {I}}} .</li></ul> <p>Even if A {\displaystyle \mathbf {A} } is not square, the two matrices A H A {\displaystyle \mathbf {A} ^{\mathrm {H} }\mathbf {A} } and A A H {\displaystyle \mathbf {A} \mathbf {A} ^{\mathrm {H} }} are both Hermitian and in fact <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive semi-definite matrices</a>. </p><p>The conjugate transpose "adjoint" matrix A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} should not be confused with the <a href="/facts/Adjugate/jjT9v8ox">adjugate</a>, adj ( A ) {\displaystyle \operatorname {adj} (\mathbf {A} )} , which is also sometimes called <i>adjoint</i>. </p><p>The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ [ a − b b a ] . {\displaystyle a+ib\equiv {\begin{bmatrix}a&-b\\b&a\end{bmatrix}}.} </p><p>That is, denoting each <i>complex</i> number z {\displaystyle z} by the <i>real</i> 2 × 2 {\displaystyle 2\times 2} matrix of the linear transformation on the <a href="/facts/Argand_diagram/vE0QrxJp">Argand diagram</a> (viewed as the <i>real</i> vector space R 2 {\displaystyle \mathbb {R} ^{2}} ), affected by complex <i> z {\displaystyle z} </i>-multiplication on C {\displaystyle \mathbb {C} } . </p><p>Thus, an m × n {\displaystyle m\times n} matrix of complex numbers could be well represented by a 2 m × 2 n {\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 n × m {\displaystyle n\times m} matrix made up of complex numbers. </p><p>For an explanation of the notation used here, we begin by representing complex numbers e i θ {\displaystyle e^{i\theta }} as the rotation matrix, that is, e i θ = ( cos θ − sin θ sin θ cos θ ) = cos θ ( 1 0 0 1 ) + sin θ ( 0 − 1 1 0 ) . {\displaystyle e^{i\theta }={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}=\cos \theta {\begin{pmatrix}1&0\\0&1\end{pmatrix}}+\sin \theta {\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} Since e i θ = cos θ + i sin θ {\displaystyle e^{i\theta }=\cos \theta +i\sin \theta } , we are led to the matrix representations of the unit numbers as 1 = ( 1 0 0 1 ) , i = ( 0 − 1 1 0 ) . {\displaystyle 1={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad i={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.} </p><p>A general complex number z = x + i y {\displaystyle z=x+iy} is then represented as z = ( x − y y x ) . {\displaystyle z={\begin{pmatrix}x&-y\\y&x\end{pmatrix}}.} The <a href="/facts/Complex_conjugate/7TEaoQDe">complex conjugate</a> operation (that sends a + b i {\displaystyle a+bi} to a − b i {\displaystyle a-bi} for real a , b {\displaystyle a,b} ) is encoded as the matrix transpose.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="properties">Properties</h2> <ul><li> ( A + B ) H = A H + B H {\displaystyle (\mathbf {A} +{\boldsymbol {B}})^{\mathrm {H} }=\mathbf {A} ^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }} for any two matrices A {\displaystyle \mathbf {A} } and B {\displaystyle {\boldsymbol {B}}} of the same dimensions.</li> <li> ( z A ) H = z ¯ A H {\displaystyle (z\mathbf {A} )^{\mathrm {H} }={\overline {z}}\mathbf {A} ^{\mathrm {H} }} for any complex number z {\displaystyle z} and any m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } .</li> <li> ( A B ) H = B H A H {\displaystyle (\mathbf {A} {\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {B}}^{\mathrm {H} }\mathbf {A} ^{\mathrm {H} }} for any m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } and any n × p {\displaystyle n\times p} matrix B {\displaystyle {\boldsymbol {B}}} . Note that the order of the factors is reversed.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a></li> <li> ( A H ) H = A {\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)^{\mathrm {H} }=\mathbf {A} } for any m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } , i.e. Hermitian transposition is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a>.</li> <li>If A {\displaystyle \mathbf {A} } is a square matrix, then det ( A H ) = det ( A ) ¯ {\displaystyle \det \left(\mathbf {A} ^{\mathrm {H} }\right)={\overline {\det \left(\mathbf {A} \right)}}} where det ( A ) {\displaystyle \operatorname {det} (A)} denotes the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of A {\displaystyle \mathbf {A} } .</li> <li>If A {\displaystyle \mathbf {A} } is a square matrix, then tr ( A H ) = tr ( A ) ¯ {\displaystyle \operatorname {tr} \left(\mathbf {A} ^{\mathrm {H} }\right)={\overline {\operatorname {tr} (\mathbf {A} )}}} where tr ( A ) {\displaystyle \operatorname {tr} (A)} denotes the <a href="/facts/Trace_(matrix)/dchFFvHt">trace</a> of A {\displaystyle \mathbf {A} } .</li> <li> A {\displaystyle \mathbf {A} } is <a href="/facts/Invertible_matrix/fPqXk3V8">invertible</a> <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} is invertible, and in that case ( A H ) − 1 = ( A − 1 ) H {\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\mathrm {H} }} .</li> <li>The <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} are the complex conjugates of the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A {\displaystyle \mathbf {A} } .</li> <li> ⟨ A x , y ⟩ m = ⟨ x , A H y ⟩ n {\displaystyle \left\langle \mathbf {A} x,y\right\rangle _{m}=\left\langle x,\mathbf {A} ^{\mathrm {H} }y\right\rangle _{n}} for any m × n {\displaystyle m\times n} matrix A {\displaystyle \mathbf {A} } , any vector in x ∈ C n {\displaystyle x\in \mathbb {C} ^{n}} and any vector y ∈ C m {\displaystyle y\in \mathbb {C} ^{m}} . Here, ⟨ ⋅ , ⋅ ⟩ m {\displaystyle \langle \cdot ,\cdot \rangle _{m}} denotes the standard complex <a href="/facts/Inner_product/HoyjElEy">inner product</a> on C m {\displaystyle \mathbb {C} ^{m}} , and similarly for ⟨ ⋅ , ⋅ ⟩ n {\displaystyle \langle \cdot ,\cdot \rangle _{n}} .</li></ul> <h2 id="generalizations">Generalizations</h2> <p>The last property given above shows that if one views A {\displaystyle \mathbf {A} } as a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> from <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> C n {\displaystyle \mathbb {C} ^{n}} to C m , {\displaystyle \mathbb {C} ^{m},} then the matrix A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} corresponds to the <a href="/facts/Hermitian_adjoint/gN6RHphd">adjoint operator</a> of 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. </p><p>Another generalization is available: suppose A {\displaystyle A} is a linear map from a complex <a href="/facts/Vector_space/M1pxMLx2">vector space</a> V {\displaystyle V} to another, W {\displaystyle W} , then the <a href="/facts/Complex_conjugate_linear_map/trWp8y4A">complex conjugate linear map</a> as well as the <a href="/facts/Transpose_of_a_linear_map/uv0JWTHq">transposed linear map</a> are defined, and we may thus take the conjugate transpose of A {\displaystyle A} to be the complex conjugate of the transpose of A {\displaystyle A} . It maps the conjugate <a href="/facts/Dual_space/4Uw4Knz1">dual</a> of W {\displaystyle W} to the conjugate dual of V {\displaystyle V} . </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Complex_dot_product/tNz8MLjT">Complex dot product</a></li> <li><a href="/facts/Hermitian_adjoint/gN6RHphd">Hermitian adjoint</a></li> <li><a href="/facts/Adjugate_matrix/jjT9v8ox">Adjugate matrix</a></li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Adjoint_matrix">"Adjoint matrix"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 2020-09-08. <a href="https://mathworld.wolfram.com/ConjugateTranspose.html" target="_blank">https://mathworld.wolfram.com/ConjugateTranspose.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Chasnov, Jeffrey R. (4 February 2022). "1.6: Matrix Representation of Complex Numbers". Applied Linear Algebra and Differential Equations. LibreTexts. <a href="https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.06%3A_Matrix_Representation_of_Complex_Numbers" target="_blank">https://math.libretexts.org/Bookshelves/Differential_Equations/Applied_Linear_Algebra_and_Differential_Equations_(Chasnov)/02%3A_II._Linear_Algebra/01%3A_Matrices/1.06%3A_Matrix_Representation_of_Complex_Numbers</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 2020-09-08. <a href="https://mathworld.wolfram.com/ConjugateTranspose.html" target="_blank">https://mathworld.wolfram.com/ConjugateTranspose.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>