Multivariate Analysis

  1. Foundations in Matrix Algebra and Random Vectors
    1. Basic Vector Operations
      1. Vector Definition and Notation
        1. Vector Addition and Subtraction
          1. Scalar Multiplication of Vectors
            1. Dot Product
              1. Vector Norms
              2. Basic Matrix Operations
                1. Matrix Definition and Notation
                  1. Matrix Addition and Subtraction
                    1. Scalar Multiplication of Matrices
                      1. Matrix Multiplication
                        1. Rules and Properties
                          1. Conformability Requirements
                          2. Matrix Transposition
                            1. Definition and Properties
                              1. Transpose of Products
                            2. Special Types of Matrices
                              1. Square Matrices
                                1. Symmetric Matrices
                                  1. Diagonal Matrices
                                    1. Identity Matrices
                                      1. Orthogonal Matrices
                                        1. Positive Definite Matrices
                                        2. Matrix Inverses
                                          1. Existence Conditions
                                            1. Calculation Methods
                                              1. Properties of Inverses
                                                1. Generalized Inverses
                                                2. Determinants
                                                  1. Definition and Calculation
                                                    1. Properties of Determinants
                                                      1. Geometric Interpretation
                                                        1. Applications in Multivariate Analysis
                                                        2. Eigenvalues and Eigenvectors
                                                          1. Definition and Interpretation
                                                            1. Characteristic Equation
                                                              1. Eigenvalue Decomposition
                                                                1. Properties and Applications
                                                                  1. Spectral Decomposition
                                                                  2. Random Vectors
                                                                    1. Definition and Notation
                                                                      1. Mean Vector
                                                                        1. Definition and Calculation
                                                                        2. Covariance Matrix
                                                                          1. Definition and Properties
                                                                            1. Sample Covariance Matrix
                                                                              1. Positive Definiteness
                                                                              2. Correlation Matrix
                                                                                1. Definition and Properties
                                                                                  1. Relationship to Covariance Matrix
                                                                                    1. Sample Correlation Matrix
                                                                                  2. Partitioned Matrices
                                                                                    1. Block Matrix Operations
                                                                                      1. Partitioned Matrix Multiplication
                                                                                        1. Inverse of Partitioned Matrices
                                                                                          1. Applications to Multivariate Statistics

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                                                                                        1. Introduction to Multivariate Analysis

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                                                                                        3. The Multivariate Normal Distribution