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 .

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