Conjugate transpose: Difference between revisions

Revision as of 16:36, 23 February 2024

Revision as of 16:46, 23 February 2024

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The conjugate transpose of a matrix <math>\mathbf{A}</math> with [[real number|real]] entries reduces to the [[transpose]] of <math>\mathbf{A}</math>, as the conjugate of a real number is the number itself.

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== Motivation ==

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>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.

Complex numbers can be usefully represented by \(2 \times 2\) real matrices, obeying matrix addition and multiplication:

\[a + ib \equiv \begin{bmatrix} a & -b \\ b & a \end{bmatrix}.\]

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

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.

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,