Matrix Group

Recently, I was working on this Undergraduate Mathematics Book , Matrix Group, which reveals the beauty of linear algebra.

At first, I was thinking of reading some books about Lie Group and Lie Algebra, that was about 3 years ago. And Lie Derivative was not easy for me to understand. Now I eventually found some key to the puzzle.

First, treat Matrix Group as some topological space equipped with a usual topology, we can define the open sets with a metric on the space. That is pretty easy. Thus, we could use our normal way to discuss the topological properties of Matrix Group, such as the compactness and connectivity.

However, it is a group, under the usual multiplication of matrices. Moreover, M_n(\mathbb{C}) is a ring. We need some restriction on Matrix Group.

A subgroup G\in GL_n(\mathbb{C}) which is also a close subspace is a matrix group over \mathbb{C}, surely we can substitute field \mathbb{C} with any other fields \mathbb{K}. Normally we call this subgroup a \mathbb{K}-matrix group.

A very impressive part is the Continuous Homomorphisms of Matrix Group, because for Matrix Group, it combines both topological properties and algebraic properties, this makes homomorphism a central role.

Let G,H be two matrix groups. A homomorphism \phi: G\rightarrow H is continuous homomorphism of matrix groups if it is continuous and its image \phi G\le H is a closed subspace of H.

We come to an example:

A map \psi:SUT_2(\mathbb{R})\rightarrow U(1),

\psi(\left[\begin{array}{cc}1&t\\ 0&1\end{array}\right])=[e^{2\pi ti}]

is a continuous surjective group homeomorphism of Matrix Group.

Here we should recall the definition, esp. the closed subspace condition. Because homeomorphisms of group do not mean of matrix group.

However, why we need to plus this restriction, it is really fantastic here. Because we shall know first, a very special function, exponential function(as well as logarithm).


which is always convergent if we treat it with Jordan blocks. This function has many good properties, the limitation of this function is the key I suppose. Well, another thing we should focus is the tangent space of Lie Algebras.

Definition: The tangent space to G at U \in G is

T_{U}G=\{\gamma '(0)\in M_{n}(\mathbb{K}),\gamma \in G, \gamma (0)=U\}

We usually use the notation \mathfrak{g}=T_{I}G, and we can assure that \mathfrak{g} has an algebraic structure as a real Lie algebra(By construction), and we wonder some topological results from \mathfrak{g}.

The exponential map: \exp, we will see that

\exp_{G}:\mathfrak{g}\rightarrow GL_{n}(\mathbb{R});\,\,\exp_{G}(X)=\exp(X),

has its image in G(how to prove it?), i.e. \exp_{G}(\mathfrak{g})\subset G, so we could just write \exp_{G}:\mathfrak{g}\rightarrow G.

  • IF G is compact and connected, then \exp_{G}(\mathfrak{g})=G.
  • THERE is an open disc N_{\mathfrak{g}}(O;r)\subset \mathfrak{g}, on which \exp is injective and gives a homemorphism(absolutely),

\exp: N_{\mathfrak{g}}(O;r)\rightarrow\exp{N_{\mathfrak{g}}(O;r)}.

Here we recall the definition of topological group and any disc around the unit is the generator of the group.


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