Lectures related to Advanced analysis II

Explores stationary points, saddle points, symmetric matrices, and orthogonal properties in optimization.

Matrix Diagonalization: Spectral Theorem

Covers the process of diagonalizing matrices, focusing on symmetric matrices and the spectral theorem.

Explores symmetric matrices, their diagonalization, and properties like eigenvalues and eigenvectors.

Symmetric Matrices: Eigenvalues and Eigenvectors

Explores the diagonalization of symmetric matrices using eigenvectors and eigenvalues, emphasizing orthogonality and real eigenvalues.

Characteristic Polynomials and Similar Matrices

Explores characteristic polynomials, similarity of matrices, and eigenvalues in linear transformations.

Implicit Function Theorem

Explores the Implicit Function Theorem, supporting hyperplanes, local extrema, and higher-order derivatives, concluding with the classification of stationary points.

Calcul de valeurs propres

Covers the calculation of eigenvalues and eigenvectors, emphasizing their significance and applications.

Diagonalization of Matrices

Explains the diagonalization of matrices, criteria, and significance of distinct eigenvalues.

Eigenvalues and Eigenvectors Decomposition

Covers the decomposition of a matrix into its eigenvalues and eigenvectors, the orthogonality of eigenvectors, and the normalization of vectors.

Non-Negative Definite Matrices and Covariance Matrices

Covers non-negative definite matrices, covariance matrices, and Principal Component Analysis for optimal dimension reduction.

Eigenvalues and Eigenvectors: Understanding Matrix Properties

Explores eigenvalues and eigenvectors, demonstrating their importance in linear algebra and their application in solving systems of equations.

Spectral Theorem Recap

Revisits the spectral theorem for symmetric matrices, emphasizing orthogonally diagonalizable properties and its equivalence with symmetric bilinear forms.

Nature of Extremum Points

Covers the nature of extremum points and their classification as stationary or saddle points.

Taylor Approximation: Extrema in Multivariable Functions

Covers Taylor approximation and extrema in multivariable functions with examples.

Symmetric Matrices: Properties and Decomposition

Covers examples of symmetric matrices and their properties, including eigenvectors and eigenvalues.

Coxeter Groups: Spectral Theorem and Sylvester's Criterion

Explores the spectral theorem, Coxeter graphs, eigenvalues, and determinants of positive definite matrices.

Symmetric Matrices: Diagonalizability and Eigenvectors

Explores the diagonalizability of symmetric matrices and their eigenvectors in an orthonormal basis.

Decomposition Spectral: Symmetric Matrices

Covers the decomposition of symmetric matrices into eigenvalues and eigenvectors.

Advanced Analysis II: Taylor Expansion

Covers the Taylor expansion and properties of the Hessian matrix for stationary points.

Eigenvalues and Similar Matrices

Explores eigenvalues, matrix trace, and similarity, highlighting their significance in matrix properties.