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Definition

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

( A H ) i j = A j i ¯ {\displaystyle \left(\mathbf {A} ^{\mathrm {H} }\right)_{ij}={\overline {\mathbf {A} _{ji}}}}

Eq.1

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.

This definition can also be written as

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

where A T {\displaystyle \mathbf {A} ^{\operatorname {T} }} denotes the transpose and 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 A {\displaystyle \mathbf {A} } can be denoted by any of these symbols:

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

In some contexts, A ∗ {\displaystyle \mathbf {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 A {\displaystyle \mathbf {A} } .

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}}}

We first transpose the matrix:

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}}}

Then we conjugate every entry of the matrix:

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}}}

Basic remarks

A square matrix A {\displaystyle \mathbf {A} } with entries a i j {\displaystyle a_{ij}} is called

• Hermitian or self-adjoint 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}}}} .
• Skew Hermitian 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}}}} .
• Normal if A H A = A A H {\displaystyle \mathbf {A} ^{\mathrm {H} }\mathbf {A} =\mathbf {A} \mathbf {A} ^{\mathrm {H} }} .
• Unitary 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}}} .

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 positive semi-definite matrices.

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

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}}.}

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

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.

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}}.}

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 complex conjugate 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.³

Properties

• ( 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.
• ( 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} } .
• ( 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.⁴
• ( 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 involution.
• 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 determinant of A {\displaystyle \mathbf {A} } .
• 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 trace of A {\displaystyle \mathbf {A} } .
• A {\displaystyle \mathbf {A} } is invertible if and only if 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} }} .
• The eigenvalues of A H {\displaystyle \mathbf {A} ^{\mathrm {H} }} are the complex conjugates of the eigenvalues of A {\displaystyle \mathbf {A} } .
• ⟨ 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 inner product on C m {\displaystyle \mathbb {C} ^{m}} , and similarly for ⟨ ⋅ , ⋅ ⟩ n {\displaystyle \langle \cdot ,\cdot \rangle _{n}} .

Generalizations

The last property given above shows that if one views A {\displaystyle \mathbf {A} } as a linear transformation from Hilbert space 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 adjoint operator 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.

Another generalization is available: suppose A {\displaystyle A} is a linear map from a complex vector space V {\displaystyle V} to another, 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 A {\displaystyle A} to be the complex conjugate of the transpose of A {\displaystyle A} . It maps the conjugate dual of W {\displaystyle W} to the conjugate dual of V {\displaystyle V} .

See also

• Complex dot product
• Hermitian adjoint
• Adjugate matrix

External links

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

References

1. Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 2020-09-08. https://mathworld.wolfram.com/ConjugateTranspose.html ↩

2. H. W. Turnbull, A. C. Aitken, "An Introduction to the Theory of Canonical Matrices," 1932. ↩

3. Chasnov, Jeffrey R. (4 February 2022). "1.6: Matrix Representation of Complex Numbers". Applied Linear Algebra and Differential Equations. LibreTexts. 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 ↩

4. Weisstein, Eric W. "Conjugate Transpose". mathworld.wolfram.com. Retrieved 2020-09-08. https://mathworld.wolfram.com/ConjugateTranspose.html ↩