This is just a really quick bunch of chapter summaries for quick reference. Hopefully I can find the time to make more insightful posts soon.
Summary of Chapter 5:
- You can exponentiate matrices by using the usual power series formula of . Everything converges and works out nicely:
- If then . Consequently so that .
- Conjugation passes through: .
- (use Jordan canonical form).
- isn’t injective except in trivial cases.
- You can rearrange absolutely summable infinite series (with terms in or even ) without altering the sum’s value; this relies on completeness.
- For with you can define the matrix logarithm by the usual power series for . For such matrices . Similar things about .
- There are neighbourhoods of and of such that is a diffeomorphism.
Summary of Chapter 6:
- One-parameter subgroup of is a continuous group homomorphism (we conflate with its image as usual).
- Every one-parameter subgroup is of the form for some , and is called the infinitesimal generator of . Show is differentiable, then let and apply matrix product rule.
Summary of Chapter 7:
- A matrix Lie group is a closed subgroup . The associated Lie algebra is .
- Define the Lie bracket by for . Then is a -vector space closed under the Lie bracket.
- This motivates us to define a matrix Lie algebra as a -vector subspace of which is closed under taking Lie brackets.
- consists of the “traceless” (trace 0) matrices.
- Other examples of the Lie algebras associated to certain matrix Lie groups.
Summary of Chapter 8:
- coordinate system: , typical definition with open covers/charts, such that the transition maps are .
- Two coordinate systems are equivalent when the transition maps are .
- A manifold consists of a topological space together with an equivalence class of coordinate systems on .
- Coordinates at identity: given by the exponential map (maps a neighbourhood of in homeomorphically to a neighbourhood of in ).
- Matrix Lie groups are analytic manifolds.
- The connected component of the identity of a matrix Lie group is an open normal subgroup generated by .