Eigenvalue problem: Eigenbasis, Spectral theorem

This lecture covers the eigenvalue problem associated with a linear operator on a vector space, defining eigenvalues and eigenvectors. It explains the matrix representation of the operator, the characteristic equation, and the algebraic multiplicity of eigenvalues. Additionally, it discusses eigenbasis, necessary conditions for its existence, and the Spectral Theorem for normal operators guaranteeing an orthonormal eigenbasis. Properties of normal operators, including spectral decomposition and orthogonality of eigenspaces, are also explored, along with the special cases of self-adjoint and unitary operators.