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1. Basics of linear algebra/quantum mechanics
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Random Matrix Theory and its Applications - Lecture 2
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- 1 Random Matrix Theory and its Applications: Recap
- 2 1. Basics of linear algebra/quantum mechanics
- 3 2. Operator
- 4 3. Orthogonal transformation
- 5 Under this orthonormal transformation
- 6 Similarity transformation
- 7 4. Eigenvectors & eigenvalues of H Hat
- 8 How does one find eigenvalues & eigenvectors?
- 9 H Hat =[Hij] -NxN matrix: Matrices with real eigenvectors
- 10 Random matrix
- 11 Define Random Matrix
- 12 Ensembles of random matrices
- 13 Wigner matrices
- 14 Example
- 15 Q: What is the joint distribution of eigenvalues?
- 16 Example: Gaussian Ensemble
- 17 Rotation invariant ensembles
- 18 Under any orthonormal transformation
- 19 Example
- 20 Exercise: Prove it for NxN case
- 21 Q&A
