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Chapter 10 - System of Equations - Debugged Codes
Algebra and geometry of equations - Matrices representing systems of equations - Row reduction, echelon form, and pivots - Gaussian elimination - Row-reduced echelon form - Gauss-Jordan elimination - Possibilities for solutions - Matrix spaces after row reduction
Chapter 11 - Matrix Determinant - Debugged Codes
Four things to know about determinants - Determinant of a 2×2 matrix - The characteristic polynomial - Determinant of a 3×3 matrix - Full procedure to compute the determinant - Determinant of a triangular matrix - Determinant and row reduction - Determinant & matrix-scalar multiplication - Determinant in theory and in practice
Chapter 12 - Matrix Inverse - Debugged Codes
Concepts and applications - Matrix inverse of a diagonal matrix - Matrix inverse for a 2×2 matrix - The MCA algorithm - Inverse via row reduction - The left inverse for tall matrices - The pseudoinverse, part 1
Chapter 13 - Projections and Orthogonalization - Debugged Codes
Projections in R2 and in RN - Orthogonal and parallel vector components - Orthogonal matrices - Orthogonalization via Gram-Schmidt - QR decomposition - Inverse via QR
Chapter 14 - Leadt-Squares - Debugged Codes
Introduction - The five steps of model-fitting - Terminology - Least-squares via left inverse - Least-squares via orthogonal projection - Least-squares via row-reduction - Model-predicted values and residuals - Least-squares example
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Chapter 15 - Eigendecomposition - Debugged Codes
What are eigenvalues and eigenvectors? - Finding eigenvalues - Finding eigenvectors - Diagonalization via eigendecomposition - Conditions for diagonalization - Distinct vs. repeated eigenvalues - Complex eigenvalues or eigenvectors - Eigendecomposition of a symmetric matrix - Eigenvalues of singular matrices - Eigenlayers of a matrix - Matrix powers and inverse - Generalized eigendecomposition
Chapter 16 - Singular Value Decomposition - Debugged Codes
Singular value decomposition - Computing the SVD - Singular values and eigenvalues - SVD of a symmetric matrix - SVD and the four subspaces - SVD and matrix rank - SVD spectral theory - SVD and low-rank approximations - Normalizing singular values - Condition number of a matrix - SVD and the matrix inverse - The MP Pseudoinverse, part 2
Chapter 17 - Quadratic Form and Definiteness - Debugged Codes
Algebraic perspective - Geometric perspective - The normalized quadratic form - Eigenvectors and quadratic form surfaces - Definiteness, geometry, and eigenvalues - The definiteness of ATA - Eigenvalues and matrix definiteness
Chapter 18 - Data and Covariance Matrices - Debugged Codes
Correlation: Motivation and interpretation - Variance and standard deviation - Covariance - Correlation coefficient - Covariance matrices - From correlation to covariance matrices
Chapter 19 - Principal Components Analysis - Debugged Codes
Interpretations and applications of PCA - How to perform a PCA - The algebraic motivation for PCA - Regularization - Is PCA always the best?
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