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Linear algebra for advanced studies
·
Module 5
Spectral Theory for Symmetric & Hermitian Operators
1
Symmetric and Hermitian Matrices
Define symmetric (real) and Hermitian (complex) matrices.
2
Real Eigenvalues of Symmetric and Hermitian Matrices
Recognize that symmetric and Hermitian matrices have real eigenvalues.
3
The Spectral Theorem and Orthonormal Eigenbases
Apply the Spectral Theorem to find an orthonormal basis of eigenvectors for any symmetric or Hermitian matrix.
4
Positive Definite and Semi-definite Matrices
Define positive definite and semi-definite matrices.
5
Eigenvalue Classification of Symmetric Matrices
Classify a symmetric matrix as positive definite, semi-definite, or indefinite using its eigenvalues.
6
Symmetric Matrices and Quadratic Forms
Relate the definiteness of a symmetric matrix to the properties of the quadratic form it defines.
Previous module
Eigenvalues, Eigenvectors, and Diagonalization
Next module
Singular Value Decomposition (SVD)