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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.
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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