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Section 2: Spectral Theory & Matrix Decompositions

Overview​

Focus on eigenvalue analysis and matrix factorizations.

Topics Covered​

Chapter 5: Eigenvalues and Eigenvectors​

  • Diagonalization (5.2)
  • Powers Aᵏ and difference equations (5.3)
  • Matrix exponentials eᴬᵗ and differential equations (5.4)
  • Complex matrices (5.5)
  • Similarity transformations (5.6)

Chapter 6: Positive Definite Matrices​

  • Tests for positive definiteness (6.2)
  • Singular Value Decomposition (6.3)
  • Minimum principles (6.4)

Appendix C: Matrix Factorizations​

Learning Objectives​

  • Master eigenvalue computations and applications
  • Understand SVD and its applications
  • Work with positive definite matrices
  • Apply spectral methods to differential equations