Symmetric matrices Click on a name to access the corresponding entry of the Glossary 1 Click on a number to access the detailed Table of Contents of the Tutorial

 MATHEMATICAL COMPLEMENTS   Cauchy-Schwarz inequality Cholesky factorization Cochran's Theorem Gauss-Markov theorem Jacobian determinant Jensen inequality Normal form of a matrix Positive definite matrix Projection matrix Quadratic forms Singular Value Decomposition Stirling's formula Symmetric matrix Wallis formula Wallis integrals

 Tutorial 1 Cauchy-Schwarz inequality for random variables.    Cauchy-Schwarz inequality for integrable functions.

 Tutorial 1 Bivariate transformations form first principles

 Tutorial 1 Demonstration of Jensen's inequality (finite and continuous).

 Tutorial 1 The full normal form of a matrix.    The reduced normal form of a matrix.

 Tutorial 1 The full Singular Value Decomposition.    The reduced Singular Value Decomposition.

 Tutorial 1 Eigenvalues and eigenvectors are real.    Eigenvectors are orthogonal.    Spectral decomposition and rank of a symmetric matrix.    Positive (semi-) definite matrices.

 Tutorial 1 Basic properties of positive definite matrices.

 Tutorial 1 Basic properties of projection matrices.    Two characteristic properties of projection matrices.    Spectral decomposition of a symmetric matrix and projections.

 Tutorial 1 Non symmetric square roots of a positive definite matrix.    Recursive Cholesky factorization.    Iterative Cholesky factorization.

 Tutorial 1 Preliminary results in Linear Algebra.    Cochran's Theorem in Linear Algebra.    Cochran's Theorem in Statistics.    Example : independence of the sample mean and the sample variance    of the normal distribution.

 Tutorial 1 Two demonstrations of Gauss-Markov theorem.    Variance of Least Squares parameters is smallest.    Mean Square Errors of predictions is smallest.

 Tutorial 1 Expectation of a quadratic form.    A necessary and sufficient condition for a quadratic form    in a multivariate normal vector to be Chi-square distributed.
 Tutorial 2 Independence of two linear forms in a multivariate     normal vector.          Independence of two Chi-square distributed quadratic forms     in a multinormal vector with arbitrary covariance matrix.

 Tutorial 1 A first crude approximation.    A (not quite complete) proof by the "saddle point" method.    A proof by the Central Limit Theorem (also incomplete).    An elementary but complete proof by Wallis formula.

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