Editing Conjugate transpose (section)

{{short description|Complex matrix A* obtained from a matrix A by transposing it and conjugating each entry}}
{{redirect|Adjoint matrix|the transpose of cofactor|Adjugate matrix}}

In [[mathematics]], the '''conjugate transpose''', also known as the '''Hermitian transpose''', of an <math>m \times n</math> [[Complex number|complex]] [[matrix (mathematics)|matrix]] <math>\mathbf{A}</math> is an <math>n \times m</math> matrix obtained by [[transpose|transposing]] <math>\mathbf{A}</math> and applying [[complex conjugate|complex conjugation]] to each entry (the complex conjugate of <math>a+ib</math> being <math>a-ib</math>, for real numbers <math>a</math> and <math>b</math>). There are several notations, such as <math>\mathbf{A}^\mathrm{H}</math> or <math>\mathbf{A}^*</math>,<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Conjugate Transpose|url=https://mathworld.wolfram.com/ConjugateTranspose.html|access-date=2020-09-08|website=mathworld.wolfram.com|language=en}}</ref> <math>\mathbf{A}'</math>,<ref>
H. W. Turnbull, A. C. Aitken,
"An Introduction to the Theory of Canonical Matrices,"
1932.
</ref> or (often in physics) <math>\mathbf{A}^{\dagger}</math>.

For [[Real number|real]] matrices, the conjugate transpose is just the transpose, <math>\mathbf{A}^\mathrm{H} = \mathbf{A}^\operatorname{T}</math>.