Applied Representation Theory : 005

Unitary Matrices

The Jacobi relation is used to revisit the relationship between the group of unit quaternions and the vector space of imaginary quaternions. This is generalized to the definition of unitary matrices.

Jacobi, Revsisted

When dealing with matrices, we have learned that

\begin{equation}\label{jacobi}\det e^{M} = e^{\mathsf{Tr}M}.\end{equation}

In particular, if \(\det e^{M} = 1\), this means that \(\mathsf{Tr} M = 0\). For complex matrices, the traceless condition can be imposed on the vector space \(\mathsf{Hom}_{n,n}\mathbb{C}\) to find a subspace:

$$\mathfrak{sl}_{n} = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{C} \;\Big|\; \mathsf{Tr}M = 0\right\}.$$

Just as \(\mathsf{Hom}_{n,n}\mathbb{C}\) is a vector space of \(\mathbb{C}\)-dimension \(n^{2}\), \(\mathfrak{sl}_{n}\) is a vector space of \(\mathbb{C}\)-dimension \(n^{2}-1\).

As we have seen in the exercises, \(\mathsf{sl}_{2}\) is three-dimensional, and we already know three linearly independent, traceless, \(2\times 2\) matrices, the \(\sigma\) matrices:

$$\sigma_{1} = \left(\begin{array}{rr}0&1\\1&0\end{array}\right),\quad\sigma_{2} = \left(\begin{array}{rr}0&-i\\i&0\end{array}\right),\quad\sigma_{3} = \left(\begin{array}{rr}1&0\\0&-1\end{array}\right).$$

Therefore, a general element of \(\mathfrak{sl}_{2}\) may be written,

$$M = a \sigma_{1} + b\sigma_{2} + c\sigma_{3},\quad a,b,c \in \mathbb{C}.$$

Therefore, any \(2\times 2\) complex matrix with unit determinant may be parameterized by these three complex numbers: \(a,b,c\).

As we have previously seen, the set of matrices with unit determinant are not a vector space, but they are closed under matrix multiplication, as

$$\det AB = \det A \det B.$$

They are also always invertible, whence, they form a group, the special linear group:

$$\mathsf{SL}_{n}\mathbb{C} = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{C} \;\Big|\; \det M = 1\right\}.$$

The relationship between the group \(\mathsf{SL}_{n}\mathbb{C}\) and the vector space \(\mathfrak{sl}_{n}\) is an example of a much deeper phenomena that will occupy the bulk of this course.

The Unit Quaternions, Revisited

We have have seen that the unit quaternions,

$$\mathbb{H}_{1} = \left\{ q \in \mathbb{H} \;\Big|\; |q| = 1\right\},$$

also have an interesting relationship with Jacobi's relation, \eqref{jacobi}. Namely,

$$\mathbb{H}_{1} \cong \left\{ e^{\Theta} \in \mathbb{H} \;\Big|\; \overline{\Theta} = -\Theta\right\}.$$

That is, the angular part of polar representation of $\mathbb{H}$ with multiplication forms a subgroup  \(\mathbb{H}_{1}\) parametrized by the vector space of purely imaginary quaternions. Since the quaternions have a known matrix representation,

$$i,j,k \rightarrow I,J,K,$$

where

$$I = i\sigma_{1},\quad J = i\sigma_{2}, \quad K = i\sigma_{3},$$

we immediately see that the vector space of imaginary quaternions:

$$\mathsf{Im}\,\mathbb{H} = \left\{ q \in \mathbb{H} \;\Big|\; q = bi + cj + dk,\quad b,c,d \in \mathbb{R}\right\},$$

is isomorphic to a linear subspace of \(\mathsf{sl}_{2}\):

$$\mathsf{Im}\,\mathbb{H} \cong \left\{ Q \in \mathsf{Hom}_{2,2}\mathbb{C} \;\Big|\; Q = bI + cJ + dK,\quad b,c,d \in \mathbb{R}\right\}\subset \mathfrak{sl}_{2}.$$

As \(\mathfrak{sl}_{2}\) has three complex dimensions, \(\mathsf{Im}\,\mathbb{H}\) has three real dimensions. Exponentiating, we find a matrix

$$ M = e^{bI + cJ + dK},$$

with a curious property that

\begin{equation}\label{unitarycondition}M^{\dagger} = \overline{M}^{\sf T} = M^{-1}.\end{equation}

Alternatively we might write

$$M^{\dagger} M = \mathbb{1}.$$

This is the condition for a unitary matrix.

Unitary Matrices

Abstracting to \(n\)-dimensions, we see the set of matrices that satisfy \eqref{unitarycondition} are themselves special. They preserve the inner product on \(\mathbb{C}^{n}\):

$$z^{\dagger}\cdot w \rightarrow (M\cdot z)^{\dagger}\cdot M\cdot w = z^{\dagger}\cdot M^{\dagger} \cdot M \cdot w = z^{\dagger}\cdot w.$$

In particular, they preserve the norm on \(\mathbb{C}^{n}\):

$$ (M\cdot z)^{\dagger}\cdot M\cdot z = z^{\dagger}\cdot M^{\dagger}\cdot M \cdot z=z^{\dagger}z$$

Transformations like \(M\) that preserve the norm\(^{1}\) of a vector space are called isometries.

These isometries are the so-called unitary matrices, 

$$\mathsf{U}_{n} = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{C} \;\Big|\; M^{\dagger} = M^{-1}\right\}.$$

The special unitary matrices are unitary matrices with unit determinant:

$$\mathsf{SU}_{n} = \left\{ M \in \mathsf{Hom}_{n,n}\mathbb{C} \;\Big|\; M^{\dagger} = M^{-1},\quad \det M = 1\right\}.$$

Taken with matrix multiplication, both sets of matrices form groups that will feature prominently in future discussions. 

Exercises

5.1 : The vector space \(\mathfrak{sl}_{2}\)

Verify that \(\mathfrak{sl}_{n}\) is a vector space. In particular, that it is a subspace of \(\mathsf{Hom}_{n,n}\mathbb{C}\) of dimension \(n-\). Show that for two matrices \(A\) and \(B\) in \(\mathsf{sl}_{n}\), 

$$A\cdot B = \mathsf{Tr} AB,$$

serves as an inner product. 

5.2 : The vector space \(\mathfrak{su}_{2}\)

The relationship between the group \(\mathbb{H}_{1}\) and the vector space \(\mathsf{Im}\,\mathbb{H}\) is very similar to that between the group \(\mathsf{SL}_{2}\mathbb{C}\) and the vector space \(\mathfrak{sl}_{2}\). For both pairs, the latter parametrizes the former via the exponential map:

$$G = e^{M} = \sum_{n = 0}^{\infty} \frac{1}{n!}M^{n}.$$

Show that \(\mathbb{H}_{1}\) is homomorphic to \(\mathsf{SU}_{2}\). Use this fact to define the real vector space \(\mathfrak{su}_{2} \cong \mathsf{Im}\,\mathbb{H} \subset \mathfrak{sl}_{2}\). Show that this definition generalizes to arbitrary \(n\), and hence define the vector space \(\mathfrak{su}_{n}\) that parametrizes \(\mathsf{SU}_{n}\).

5.3 : Unitary determinants

Because a unitary matrix \(U\in\mathsf{U}_{n}\) satisfies

$$U^{\dagger}U = \mathbb{1},$$

its determinant also satisfies

$$|\det U|^{2} = 1.$$

Therefore, \(\det U\) is a member of \(\mathsf{U}_{1}\), the unit complex numbers. Just as you showed an arbitrary orthogonal matrix in \(\mathsf{O}_{n}\) is the product of a rotation and a reflection, so that a general \(U\) in \(\mathsf{U}_{n}\) is the product of a member of \(\mathsf{SU}_{n}\) and some phase in \(\mathsf{U}_{1}\). Hence argue

$$\mathsf{U}_{n}\cong \mathsf{SU}_{n}\times \mathsf{U}_{1}.$$

What is the corresponding group \(G\) in 

$$\mathsf{O}_{n}\cong \mathsf{SO}_{n}\times G?$$

5.4 : The vector space \(\mathfrak{so}_{n}\)

 Define the vector space \(\mathfrak{so}_{n}\) associated with \(\mathsf{SO}_{n}\) in the same way that \(\mathfrak{su}_{n}\) is associated with \(\mathsf{SU}_{n}\).

5.5 : Relationships between \(\mathfrak{so}\) and \(\mathfrak{su}\)

Show that \(\mathfrak{so}_{3} \cong \mathfrak{su}_{2}\). Hence show that  \(\mathfrak{so}_{4} \cong \mathfrak{su}_{2}\times\mathsf{su}_{2} \cong \mathfrak{sl}_{2}\). 

5.6 : The vector space \(\mathfrak{sl}_{3}\)

 Find an explicit basis for the space $\mathfrak{sl}_{3}$.

5.7 : The inner automorphisms of \(\mathsf{GL}_{n}\mathbb{C}\)

Conjugation of a matrix \(M\) by a matrix \(U\) is defined by the map \(\psi_{U}\):

\begin{equation}\label{conjg}\psi_{U} : M \rightarrow U\cdot M \cdot U^{-1}.\end{equation}

Under what condition is \(\psi_{U}\) is a linear transformation? For a matrix group \(G\), define the set of (inner) automorphisms of \(G\) as \(\Psi_{G}\):

$$\Psi_{G} = \left\{\psi_{U} \;\Big|\; U \in G\right\},$$

where the conjugation map \(\psi_{U}\) is defined as in \eqref{conjg}, with \(M\) assumed to be in \(G\). If \(U\) in  \(\mathsf{SO}_{3}\), argue that \(\psi_{U}\) amounts a passive rotation, that is, a change in reference frame. (Hint: think of \(M\) as an active rotation.)

Define a multiplication operation \(\circ\) on \(\Psi_{G}\) by composition. That is, for a matrix \(M\):

$$\psi_{U}\circ\psi_{V}(M) = \psi_{U}(\psi_{V}(M)) = V\cdot U \cdot M \cdot U^{-1} \cdot V^{-1}.$$

Show that \((\Psi_{G},\circ)\) is a group that is homomorphic to \(G\). 

5.8 : The Double Cover of \(\mathsf{SO}_{3}\)

Let \(G = \mathsf{SU}_{2}\cong \mathbb{H}_{1}\). Show that the elements of \(\Psi_{G}\) can also act as a linear transformation of the vector space \(\mathsf{Im}\,\mathbb{H}\cong \mathfrak{su}_{2}\). In particular, show that \(\Psi_{G}\) preserves the norm on \(\mathfrak{su}_{2}\). Argue that \(\Psi_{g}\) is homomorphic to \(\mathsf{SO}_{3}\). Hence show that each element in \(\mathsf{SO}_{3}\) corresponds to two elements of \(\mathsf{SU}_{2}\). That is, show that \(\mathsf{SU}_{2}\) is a double cover of \(\mathsf{SO}_{3}\).

\(^{1}\) More precisely, if that norm is used to define a notion of distance on that vector space. This has natural generalization to curved surfaces and other manifolds. Often in applications this is framed in terms of preserving a Riemannian metric.

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