Matrix Groups and Lie Theory

Group Definition

A group is a set \(G\) with an operation "\(\circ\)" on the elmenets of \(G\) such that it satisfies:

1. Closure: \(\forall g_1, g_2 \in G, g_1 \circ g_2 \in G\).
2. Associativity: \(\forall g_1, g_2, g_3 \in G, g_1 \circ (g_2 \circ g_3) = (g_1 \circ g_2) \circ g_3 = g_1 \circ g_2 \circ g_3\).
3. Identity: \(\exists g_0 \in G, \ \text{s.t.} \ \forall g \in G, g_0 \circ g = g \cdot g_0 = g\).
4. Invertibility: \(\forall g \in G, \exists g^{-1} \in G, \ \text{s.t.} \ g \circ g^{-1} = g_0\).

An example group is \(G = (\mathbb{Z}, +)\).

Matrix Groups

Group Relationships

\[ \begin{align*} SO(n) \subset O(n) \subset GL(n), \quad SE(n) \subset E(n) \subset A(n) \subset GL(n + 1) \end{align*} \]

Group Name	Definition
General Linear Group, \(GL(n)\)	Let \(\mathcal{M}(n)\) be a set of all real \(n \times n\) matrices. The general linear group, \(GL(n)\), consists of all \(\mathbf{A} \in \mathcal{M}(n)\) for which \(\text{det}(\mathbf{A}) \neq 0.\) In order words, the set of all \(n \times n\) non-singular (real) matrices with matrix multiplication is \(GL(n)\).
Special Linear Group, \(SL(n)\)	All matrices \(\mathbf{A} \in GL(n)\) for which \(\text{det}(\mathbf{A}) = 1\) form a group called the special linear group \(SL(n)\). Note that the inverse of \(\mathbf{A}\) is also in this group since \(\text{det}(\mathbf{A}^{-1}) = \text{det}(\mathbf{A})^{-1}\).
Affine Group, \(A(n)\)	An affine transformation \(L \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^n\) is defined jointly by a matrix \(\mathbf{A} \in GL(n)\) and a vector \(\mathbf{b} \in \mathbb{R}^n\) such that \(L(\mathbf{x}) = \mathbf{A} \mathbf{x} + \mathbf{b}.\) The set of all such affine transformations is called the affine group of dimension \(n\) and is denoted by \(A(n)\). Note that \(L\) defined above is not a linear map unless \(\mathbf{b} = \mathbf{0}\).
Orthogonal Group, \(O(n)\)	A matrix \(\mathbf{A} \in \mathcal{M}(n)\) is called orthogonal if it preserves the inner product, i.e. \(\langle \mathbf{A} \mathbf{x}, \mathbf{A} \mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{y} \rangle, \quad \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n.\) The set of all orthogonal matrices forms the orthogonal group \(O(n)\), which is a subgroup of \(GL(n)\). For an orthogonal matrix \(\mathbf{R}\), we have \(\langle \mathbf{R} \mathbf{x}, \mathbf{R} \mathbf{y} \rangle = \mathbf{x}^T \mathbf{R}^T \mathbf{R} \mathbf{y} = \mathbf{x}^T \mathbf{y}, \quad \forall \mathbf{x}, \mathbf{y} \in \mathbb{R}^n,\) which suggests that \(\mathbf{R}^T \mathbf{R} = \mathbf{R} \mathbf{R}^T = \mathbf{I}\), or equivalently \(O(n) = \left\{ \mathbf{R} \in GL(n) \ \| \ \mathbf{R}^T \mathbf{R} = \mathbf{I} \right\}.\) The above shows that for any orthogonal matrix \(\mathbf{R}\), we have \(\text{det}(\mathbf{R}^T \mathbf{R}) = \left(\text{det}(\mathbf{R}) \right)^2 = \text{det}(\mathbf{I}) = 1\), such that \(\text{det}(\mathbf{R}) \in \left\{ \pm 1 \right\}\).
Special Orthogonal Group, \(SO(3)\)	The subgroup of \(O(n)\) with \(\text{det}(\mathbf{R}) = +1\) is called the special orthogonal group \(SO(n)\). Note that \(SO(n) = O(n) \cap SL(n)\). \(SO(3)\) is a Lie group and represents rotations.
Euclidean Group, \(E(n)\)	A Euclidean transformation \(L\) from \(\mathbb{R}^n \rightarrow \mathbb{R}^n\) is defined by an orthogonal matrix \(\mathbf{R} \in O(n)\) and a vector \(\mathbf{t} \in \mathbf{R}^n\) such that: \(L \ : \ \mathbb{R}^n \rightarrow \mathbb{R}^n; \quad \mathbf{x} \rightarrow \mathbf{R} \mathbf{x} + \mathbf{t}\). The set of all such transformations is called the Euclidean group \(E(n)\). It is a subgroup of the affine group \(A(n)\). Embedded by homogeneous coordinates, we have \(E(n) = \left\{\left(\begin{array}{cc}\mathbf{R} & \mathbf{t} \\\mathbf{0} & 1\end{array}\right) \ \|\ \mathbf{R} \in O(n), \mathbf{t} \in \mathbb{R}^n\right\}.\)
Special Euclidean Group, \(SE(n)\)	For \(E(n)\), if \(\mathbf{R} \in SO(n)\), we have the special Euclidean group \(SE(n)\). \(SE(3)\) is a Lie group and represents poses.

Lie Group and Lie Algebra

Lie group refer to a group with smooth manifold properties. Smoothness implies that we can use differential calculus on the manifold; or roughly, if we change the input to any group operation by a little bit, the output will only change by a little bit. With every matrix Lie group is associated a Lie algebra, which consists of a vector space \(\mathbb{V}\) over some field \(\mathbb{F}\), and a binary operation \(\left[\cdot, \cdot \right]\), called the Lie bracket. The following properties are satisfied for Lie Algebra \((\mathbb{V}, \mathbb{F}, \left[\cdot, \cdot\right])\):

1. Closure: \(\forall \mathbf{X}, \mathbf{Y} \in \mathbb{V}; \ \left[ \mathbf{X}, \mathbf{Y} \right] \in \mathbb{V}\).

2. Bilinearity: \(\forall \mathbf{X}, \mathbf{Y}, \mathbf{Z} \in \mathbb{V}; \ a,b \in \mathbb{F}\), we have:

   \[ \begin{align} &\left[a \mathbf{X} + b \mathbf{Y}, \mathbf{Z} \right] = a \left[\mathbf{X}, \mathbf{Z} \right] + b \left[ \mathbf{Y}, \mathbf{Z} \right] \\ &\left[\mathbf{Z}, a \mathbf{X} + b \mathbf{Y} \right] = a \left[ \mathbf{Z}, \mathbf{X} \right] + b \left[ \mathbf{Z}, \mathbf{Y} \right]. \end{align} \]

3. Alternating: \(\forall \mathbf{X} \in \mathbb{V}; \ \left[ \mathbf{X}, \mathbf{X} \right] = 0\).

4. Jacobi identity: \(\forall \mathbf{X}, \mathbf{Y}, \mathbf{Z} \in \mathbb{V}; \ \left[ \mathbf{X}, \left[ \mathbf{Y}, \mathbf{Z} \right] \right] + \left[ \mathbf{Z}, \left[ \mathbf{X}, \mathbf{Y} \right] \right] + \left[ \mathbf{Y}, \left[ \mathbf{Z}, \mathbf{X} \right] \right] = 0\).

The vector space of a Lie algebra is the tangent space of the associated Lie group at the identity element of the group, and it completely captures the local structure of the group.

References

1. Barfoot, T., State Estimation for Robotics
2. Ma., Y., An Invitation to 3-D Vision: From Images to Models, 2001