# Conjugate transpose

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{{#invoke:Hatnote|hatnote}}Template:Main other In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, bedaggered matrix, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry (i.e., negating their imaginary parts but not their real parts). The conjugate transpose is formally defined by

${\displaystyle ({\mathbf {A} }^{*})_{ij}={\overline {{\mathbf {A} }_{ji}}}}$

where the subscripts denote the i,j-th entry, for 1 ≤ in and 1 ≤ jm, and the overbar denotes a scalar complex conjugate. (The complex conjugate of ${\displaystyle a+bi}$, where a and b are reals, is ${\displaystyle a-bi}$.)

This definition can also be written as

${\displaystyle \mathbf {A} ^{*}=({\overline {\mathbf {A} }})^{\mathrm {T} }={\overline {\mathbf {A} ^{\mathrm {T} }}}}$

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

Other names for the conjugate transpose of a matrix are Hermitian conjugate, or transjugate. The conjugate transpose of a matrix A can be denoted by any of these symbols:

In some contexts, ${\displaystyle {\mathbf {A} }^{*}\,\!}$ denotes the matrix with complex conjugated entries, and the conjugate transpose is then denoted by ${\displaystyle {\mathbf {A} }^{*{\mathrm {T} }}\,\!}$ or ${\displaystyle {\mathbf {A} }^{{\mathrm {T} }*}\,\!}$.

## Example

If

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

then

${\displaystyle \mathbf {A} ^{*}={\begin{bmatrix}1&1-i\\-2+i&-i\end{bmatrix}}}$

## Basic remarks

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

Even if A is not square, the two matrices AA and AA are both Hermitian and in fact positive semi-definite matrices.

Finding the conjugate transpose of a matrix A with real entries reduces to finding the transpose of 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:

${\displaystyle a+ib\equiv \left({\begin{matrix}a&-b\\b&a\end{matrix}}\right).}$

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 ${\displaystyle \mathbb {R} ^{2}}$) affected by complex z-multiplication on ${\displaystyle \mathbb {C} }$.

An m-by-n matrix of complex numbers could therefore equally well be 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.

## Generalizations

The last property given above shows that if one views A as a linear transformation from Euclidean Hilbert space Cn to Cm, then the matrix A corresponds to the adjoint operator of 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.