Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry
In mathematics, the conjugate transpose (or Hermitian transpose) of an m-by-n matrix ${\boldsymbol {A}}$ with complex entries, is the n-by-m matrix obtained from ${\boldsymbol {A}}$ by taking the transpose and then taking the complex conjugate of each entry (the complex conjugate of $a+ib$ being $a-ib$, for real numbers $a$ and $b$). It is often denoted as ${\boldsymbol {A}}^{\mathrm {H} }$ or ${\boldsymbol {A}}^{*}$.^{[1]}^{[2]}^{[3]}
For real matrices, the conjugate transpose is just the transpose, ${\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{\mathsf {T}}$.
Definition
The conjugate transpose of an $m\times n$ matrix ${\boldsymbol {A}}$ is formally defined by
where the subscripts denote the $(i,j)$-th entry, for $1\leq i\leq n$ and $1\leq j\leq m$, and the overbar denotes a scalar complex conjugate.
This definition can also be written as^{[3]}
- ${\boldsymbol {A}}^{\mathrm {H} }=\left({\overline {\boldsymbol {A}}}\right)^{\mathsf {T}}={\overline {{\boldsymbol {A}}^{\mathsf {T}}}}$
where ${\boldsymbol {A}}^{\mathsf {T}}$ denotes the transpose and ${\overline {\boldsymbol {A}}}$ 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 ${\boldsymbol {A}}$ can be denoted by any of these symbols:
- ${\boldsymbol {A}}^{*}$, commonly used in linear algebra^{[3]}
- ${\boldsymbol {A}}^{\mathrm {H} }$, commonly used in linear algebra^{[1]}
- ${\boldsymbol {A}}^{\dagger }$ (sometimes pronounced as A dagger), commonly used in quantum mechanics
- ${\boldsymbol {A}}^{+}$, although this symbol is more commonly used for the Moore-Penrose pseudoinverse
In some contexts, ${\boldsymbol {A}}^{*}$ denotes the matrix with only complex conjugated entries and no transposition.
Example
Suppose we want to calculate the conjugate transpose of the following matrix ${\boldsymbol {A}}$.
- ${\boldsymbol {A}}={\begin{bmatrix}1&-2-i&5\\1+i&i&4-2i\end{bmatrix}}$
We first transpose the matrix:
- ${\boldsymbol {A}}^{\mathsf {T}}={\begin{bmatrix}1&1+i\\-2-i&i\\5&4-2i\end{bmatrix}}$
Then we conjugate every entry of the matrix:
- ${\boldsymbol {A}}^{\mathrm {H} }={\begin{bmatrix}1&1-i\\-2+i&-i\\5&4+2i\end{bmatrix}}$
A square matrix ${\boldsymbol {A}}$ with entries $a_{ij}$ is called
- Hermitian or self-adjoint if ${\boldsymbol {A}}={\boldsymbol {A}}^{\mathrm {H} }$; i.e., $a_{ij}={\overline {a_{ji}}}$ .
- Skew Hermitian or antihermitian if ${\boldsymbol {A}}=-{\boldsymbol {A}}^{\mathrm {H} }$; i.e., $a_{ij}=-{\overline {a_{ji}}}$ .
- Normal if ${\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }$.
- Unitary if ${\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {A}}^{-1}$, equivalently ${\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }={\boldsymbol {I}}$, equivalently ${\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}={\boldsymbol {I}}$.
Even if ${\boldsymbol {A}}$ is not square, the two matrices ${\boldsymbol {A}}^{\mathrm {H} }{\boldsymbol {A}}$ and ${\boldsymbol {A}}{\boldsymbol {A}}^{\mathrm {H} }$ are both Hermitian and in fact positive semi-definite matrices.
The conjugate transpose "adjoint" matrix ${\boldsymbol {A}}^{\mathrm {H} }$ should not be confused with the adjugate, $\operatorname {adj} ({\boldsymbol {A}})$, which is also sometimes called adjoint.
The conjugate transpose of a matrix ${\boldsymbol {A}}$ with real entries reduces to the transpose of ${\boldsymbol {A}}$, 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:
- $a+ib\equiv {\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.$
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 $\mathbb {R} ^{2}$), affected by complex z-multiplication on $\mathbb {C}$.
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
- $({\boldsymbol {A}}+{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {A}}^{\mathrm {H} }+{\boldsymbol {B}}^{\mathrm {H} }$ for any two matrices ${\boldsymbol {A}}$ and ${\boldsymbol {B}}$ of the same dimensions.
- $(z{\boldsymbol {A}})^{\mathrm {H} }={\overline {z}}{\boldsymbol {A}}^{\mathrm {H} }$ for any complex number $z$ and any m-by-n matrix ${\boldsymbol {A}}$.
- $({\boldsymbol {A}}{\boldsymbol {B}})^{\mathrm {H} }={\boldsymbol {B}}^{\mathrm {H} }{\boldsymbol {A}}^{\mathrm {H} }$ for any m-by-n matrix ${\boldsymbol {A}}$ and any n-by-p matrix ${\boldsymbol {B}}$. Note that the order of the factors is reversed.^{[2]}
- $({\boldsymbol {A}}^{\mathrm {H} })^{\mathrm {H} }={\boldsymbol {A}}$ for any m-by-n matrix ${\boldsymbol {A}}$, i.e. Hermitian transposition is an involution.
- If ${\boldsymbol {A}}$ is a square matrix, then $\operatorname {det} ({\boldsymbol {A}}^{\mathrm {H} })={\overline {\operatorname {det} ({\boldsymbol {A}})}}$ where $\operatorname {det} (A)$ denotes the determinant of ${\boldsymbol {A}}$ .
- If ${\boldsymbol {A}}$ is a square matrix, then $\operatorname {tr} ({\boldsymbol {A}}^{\mathrm {H} })={\overline {\operatorname {tr} ({\boldsymbol {A}})}}$ where $\operatorname {tr} (A)$ denotes the trace of ${\boldsymbol {A}}$.
- ${\boldsymbol {A}}$ is invertible if and only if ${\boldsymbol {A}}^{\mathrm {H} }$ is invertible, and in that case $({\boldsymbol {A}}^{\mathrm {H} })^{-1}=({\boldsymbol {A}}^{-1})^{\mathrm {H} }$.
- The eigenvalues of ${\boldsymbol {A}}^{\mathrm {H} }$ are the complex conjugates of the eigenvalues of ${\boldsymbol {A}}$.
- $\langle {\boldsymbol {A}}x,y\rangle _{m}=\langle x,{\boldsymbol {A}}^{\mathrm {H} }y\rangle _{n}$ for any m-by-n matrix ${\boldsymbol {A}}$, any vector in $x\in \mathbb {C} ^{n}$ and any vector $y\in \mathbb {C} ^{m}$. Here, $\langle \cdot ,\cdot \rangle _{m}$ denotes the standard complex inner product on $\mathbb {C} ^{m}$, and similarly for $\langle \cdot ,\cdot \rangle _{n}$.
Generalizations
The last property given above shows that if one views ${\boldsymbol {A}}$ as a linear transformation from Hilbert space $\mathbb {C} ^{n}$ to $\mathbb {C} ^{m},$ then the matrix ${\boldsymbol {A}}^{\mathrm {H} }$ corresponds to the adjoint operator of ${\boldsymbol {A}}$. 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 $A$ 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 $V$.
See also
References
External links