Symmetric matrices

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MATHEMATICAL COMPLEMENTS

 

 

 

 

 

 

Cauchy-Schwarz inequality

 

Tutorial

1

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

 

 

 

Jacobian determinant

 

Tutorial

1

Bivariate transformations form first principles

 

 

 

Jensen's inequality

 

Tutorial

1

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

 

 

 

Normal form of a matrix

 

 

Tutorial

1

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

 

 

 

Singular Value Decomposition

 

 

Tutorial

1

The full Singular Value Decomposition.
   The reduced Singular Value Decomposition.

 

 

 

Symmetric matrix

 

 

Tutorial

1

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

 

 

 

Positive definite matrix

 

 

Tutorial

1

Basic properties of positive definite matrices.


 

 

 

Projection matrix

 

 

Tutorial

1

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

 

 

 

Cholesky factorization

 

Tutorial

1

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

  

 

 

Cochran's theorem

 

 

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.

 

 

 

Gauss-Markov theorem

 

Tutorial

1

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

  

 

 

Quadratic forms

  

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.

 

 

 

Stirling's formula

 

 

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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