Matrix Theory

Theoretical Concepts

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• The Geometry of Action: Vector Spaces and the Transformations That Shape Them
• Eigenvalues & Eigenvectors: The Quest for Simplicity in Linear Transformations
• Matrix Similarity: Unveiling Invariant Structures
• The Geometry of Linear Transformations (Part 1)
• The Geometry of Linear Transformations (Part 2)
• Geometric Decompositions: Rotation, Reflection, and Stretch
• Computational Decompositions: The Triangular Factorizations
• Theoretical Decompositions: The Schur and Jordan Forms
• The Measure of Matrices: Norms, Condition Numbers, and Sensitivity
• Beyond Scalars: An Introduction to Matrix Functions
• Matrix Calculus: The Language of Optimization
• Perturbation Theory and Pseudospectra: The Stability of Matrices
• Random Matrix Theory: Finding Structure in Large-Dimensional Systems

Computational Aspects

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• Foundations of Matrix Computation: Cost, the Stack, and the LU Decomposition
• Orthogonality and Structure: The QR and Cholesky Decompositions
• The Least Squares Problem and the Crucial Role of Numerical Stability
• Numerical Computation of Eigenvalues and Eigenvectors
• The SVD in Action: From Calculation to Data Compression and Analysis
• Matrix Calculus for Gradient-Based Optimization