Random Matrix Theory and its Applications - Lecture 2

Delve into the second lecture of Random Matrix Theory and its Applications, presented by Satya Majumdar at the Bangalore School on Statistical Physics - X. Explore key concepts in linear algebra and quantum mechanics, including operators, orthogonal transformations, and eigenvectors. Learn about random matrices, their ensembles, and the Gaussian Ensemble. Examine rotation-invariant ensembles and their properties. Engage with examples, exercises, and a Q&A session to deepen your understanding of this advanced topic in statistical physics.

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