<aside> 💡 This is a personal review note. Most of the explanations come from textbooks or online resources, especially the classic vSLAM book: Basic Knowledge on Visual SLAM: From Theory to Practice. For further details, please refer to the References section at the end of this page. if you need more information, please leave a comment.
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<aside> 💡 A matrix group is a group of invertible matrices.
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A group is a non-empty set $G$ together with a binary operation on $G$, here denoted "$\cdot$", that combines any two elements $a$ and $b$ of $G$ to form an element of $G$, denoted $a\cdot b$,
$$ \forall a_1,a_2 \in A, \; a_1 \cdot a_2 \in A $$
such that the following three requirements, known as group axioms, are satisfied:
Associativity property:
$$ \forall a,b,c\in A, \; (a\cdot b)\cdot c= a \cdot (b \cdot c) $$
Identity (Unit) element
$$ \exists e \in A, \; \text{s.t.} \forall a \in A, \; a \cdot e = e \cdot a = a $$
Inverse element
$$ \forall a \in A, \exist a^{-1} \in A,\; \text{s.t.} \; a \cdot a^{-1} = e. $$
where $e$ is the Identity element
<aside> 💡 Lie Group refers to a group with continuous (smooth) properties.
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A Lie algebra consists of a set $\mathbb{V}$, a scalar field $\mathbb{F}$, and a binary operation $[\cdot , \cdot]$ called Lie bracket. If they satisfy the following properties, then $\mathbb{(V, F, [\cdot,\cdot ])}$ is a Lie algebra, denoted as $\mathfrak{g}$.
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Bilinearity
$$ \forall \bold{X,Y,Z} \in \mathbb{V};a,b \in \mathbb{F} \\ [a\bold{X}+b\bold{Y},\bold{Z}]=a[\bold{X},\bold{Z}]+b[\bold{Y},\bold{Z}] \\ [\bold{Z},a\bold{X}+b\bold{Y}]=a[\bold{Z},\bold{X}]+b[\bold{Z},\bold{Y}] $$
Alternativity (Reflexive)
$$ \forall\bold{X}\in \mathbb{V}; [\bold{X},\bold{X}]=\bold{0} $$