Conjugate transpose: Difference between revisions

Revision as of 17:52, 15 May 2025

Latest revision as of 02:17, 10 August 2025

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The conjugate transpose "adjoint" matrix <math>\mathbf{A}^\mathrm{H}</math> should not be confused with the [[adjugate]], <math>\operatorname{adj}(\mathbf{A})</math>, which is also sometimes called ''adjoint''.

The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by <math>2 \times 2</math> real matrices, obeying [[matrix addition]] and multiplication:

<math display="block">a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.</math>

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

For an explanation of the notation used here, we begin by representing complex numbers <math>e^{i\theta}</math> as the [[rotation matrix]], that is,

<math display="block">

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}.