Learn · An Introduction to Matrix Lie Groups
Lecture 14 of 59Week 4
Inner products
Prof. Krishnendu Gongopadhyay · IISER Mohali
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- From
- An Introduction to Matrix Lie Groups, IISER Mohali (at NPTEL)
- Licence
- CC BY-NC-SA, as NPTEL states it. Non-commercial use only.
- Attribution
- NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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- Tweens profiles and older. How courses are placed
All the lectures
- Complex Numbers
- The Plane Rotations
- The Quaternions
- Quaternionic matrices
- What is a Matrix Lie Group
- General Linear Group
- Complex matrices as real matrices
- Quaternionic matrices as real matrices
- Discussions on quaternions
- Topology of matrix groups
- Topology of matrix groups: Closedness
- Path-Connected subsets in R^m
- Compactness in R^m
- Inner products
- Unitary transformations
- Unitary matrices
- Congruence
- Euclidean isometries
- Two dimensional Euclidean Isometries
- Compactness of unitary groups
- Connectedness of complex matrix groups
- Connectedness of real matrix groups
- Differentiability of a function
- Differentiability of a function-II
- What is a Lie Algebra
- Examples of Lie algebras
- Lie algebra for special linear groups
- Lie algebra as vector fields
- Lie algebras for orthogonal and unitary groups
- Series of Matrices
- Series of Matrices
- Exponential map
- Properties of Exponential Map
- One parameter Subgroups
- Matrix logarithm
- Inverse Function Theorem
- Image of the Exponential map.
- Exponential map is a local diffeomorphism
- Every matrix group is a manifold
- Calculus on Manifolds
- The Lie Bracket
- Lie Algebra isomorphisms and Matrix Group isomorphisms
- Example of isomorphic Lie algebras
- The adjoint Representation
- Adjoint representation for compact matrix groups
- Shape of space rotations
- More double covers
- More double covers-II
- Overview of Simply Connectedness
- Correspondence between connected subgroups and Lie Ideals
- Discrete subgroups - I
- Discrete subgroups - II
- Characterization of Torus
- Maximal Torus
- Non-compact Examples: Lorentzian groups
- Symplectic groups
- Introduction to Smooth Manifolds
- Tangent spaces to Smooth Manifolds
- Lie groups