Unitary Transformations

Unitary transformations are analogous to rotations, but over the complex numbers, rather than the reals.

We start again in two dimensions. Let $v$ be a complex vector \begin{equation} v = \begin{pmatrix} w\\ z\\ \end{pmatrix} \end{equation} that is, a vector with complex components $w,z\in\CC$; we write this as $v\in\CC^2$. The squared length of $v$ is given by \begin{equation} |v|^2 = |w|^2 + |z|^2 = \bar{w}w + \bar{z}z = v^\dagger v \end{equation} where $v^\dagger$ denotes the Hermitian conjugate of $v$, defined by \begin{equation} v^\dagger = \begin{pmatrix} \bar{w} & \bar{z}\\ \end{pmatrix} \end{equation} or in other words the complex conjugate of the transpose of $v$. The unitary group $\Ug(2)$ consists of those complex matrices $M$ that preserve the length of $v$, that is, for which \begin{equation} v^\dagger v = (Mv)^\dagger (Mv) = v^\dagger M^\dagger M v \end{equation} for any $v\in\CC^2$, which can only be true if \begin{equation} M^\dagger M = I \end{equation} Since \begin{equation} \det(M^\dagger) = \bar{\det(M)} \end{equation} we must have \begin{equation} |\det M| = 1 \end{equation} and it is straightforward to generalize this construction to higher dimensions. The unitary groups can therefore be expressed as \begin{align} \Ug(n) &= \{ M\in\CC^{n\times n} : M^\dagger M = I \} \\ \SU(n) &= \{ M\in\CC^{n\times n} : M^\dagger M = I, \det M = 1 \} \end{align} which are often taken as the definitions of the unitary and special unitary groups, respectively. The dimension of the unitary groups is given by \begin{equation} |\SU(n)| = n^2 - 1 \end{equation}

In the remainder of this chapter, we set the stage for later developments by discussing the properties of several important unitary groups. In certain dimensions, unitary transformations can also be expressed in terms of division algebras other than $\CC$; we save that discussion for §9.2.

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