Complex Transpose

In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an $m \times n$ complex matrix $\boldsymbol$ is an $n \times m$ matrix obtained by transposing $\boldsymbol$ and applying complex conjugate on each entry (the complex conjugate of $a+ib$ being $a-ib$, for real numbers $a$ and $b$). It is often denoted as $\boldsymbol^\mathrm$ or $\boldsymbol^*$
or
$\boldsymbol'$.
H. W. Turnbull, A. C. Aitken,
"An Introduction to the Theory of Canonical Matrices,"
1932.

For real matrices, the conjugate transpose is just the transpose, $\boldsymbol^\mathrm = \boldsymbol^\mathsf$.

Definition

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

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

This definition can also be written as
:$\boldsymbol^\mathrm = \left(\overline\right)^\mathsf = \overline$

where $\boldsymbol^\mathsf$ denotes the transpose and $\overline$ denotes the matrix with complex conjugated entries.

Other names for the conjugate transpose of a matrix are Hermitian conjugate, adjoint matrix or transjugate. The conjugate transpose of a matrix $\boldsymbol$ can be denoted by any of these symbols:
* $\boldsymbol^*$, commonly used in linear algebra
* $\boldsymbol^\mathrm$, commonly used in linear algebra
* $\boldsymbol^\dagger$ (sometimes pronounced as ''A dagger''), commonly used in quantum mechanics
* $\boldsymbol^+$, although this symbol is more commonly used for the Moore–Penrose pseudoinverse

In some contexts, $\boldsymbol^*$ denotes the matrix with only complex conjugated entries and no transposition.

Example

Suppose we want to calculate the conjugate transpose of the following matrix $\boldsymbol$.
:$\boldsymbol = \begin 1 & -2 - i & 5 \\ 1 + i & i & 4-2i \end$
We first transpose the matrix:
:$\boldsymbol^\mathsf = \begin 1 & 1 + i \\ -2 - i & i \\ 5 & 4-2i\end$
Then we conjugate every entry of the matrix:
:$\boldsymbol^\mathrm = \begin 1 & 1 - i \\ -2 + i & -i \\ 5 & 4+2i\end$

Basic remarks

A square matrix $\boldsymbol$ with entries $a_$ is called
* Hermitian or self-adjoint if $\boldsymbol=\boldsymbol^\mathrm$; i.e., $a_ = \overline$.
* Skew Hermitian or antihermitian if $\boldsymbol=-\boldsymbol^\mathrm$; i.e., $a_ = -\overline$.
* Normal if $\boldsymbol^\mathrm \boldsymbol = \boldsymbol \boldsymbol^\mathrm$.
* Unitary if $\boldsymbol^\mathrm = \boldsymbol^$, equivalently $\boldsymbol\boldsymbol^\mathrm = \boldsymbol$, equivalently $\boldsymbol^\mathrm\boldsymbol = \boldsymbol$.

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

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

The conjugate transpose of a matrix $\boldsymbol$ with real entries reduces to the transpose of $\boldsymbol$, 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 $2 \times 2$ real matrices, obeying matrix addition and multiplication:
:$a + ib \equiv \begin a & -b \\ b & a \end.$

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

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

Properties of the conjugate transpose

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

Generalizations

The last property given above shows that if one views $\boldsymbol$ as a linear transformation from Hilbert space $\mathbb^n$ to $\mathbb^m ,$ then the matrix $\boldsymbol^\mathrm$ 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

* Complex dot product
*Hermitian adjoint
*Adjugate matrix

References

External links

* {{springer, title=Adjoint matrix, id=p/a010850

Linear algebra
Matrices