Lectures related to Quadratic Forms and Hecke Operators

Matrices and Quadratic Forms: Key Concepts in Linear Algebra

Provides an overview of symmetric matrices, quadratic forms, and their applications in linear algebra and analysis.

Explores symmetric matrices, diagonalization, and quadratic forms properties.

Multivariate Statistics: Wishart and Hotelling T²

Explores the Wishart distribution, properties of Wishart matrices, and the Hotelling T² distribution, including the two-sample Hotelling T² statistic.

Symmetric Matrices and Quadratic Forms

Explores symmetric matrices, quadratic forms, diagonalization, and definiteness with examples and calculations.

Characteristic Polynomials and Similar Matrices

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

Orthogonal Projection Theorems

Covers the theorems related to orthogonal projection and orthonormal bases.

Symmetric Matrices: Diagonalization

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

Singular Value Decomposition: Applications and Interpretation

Explains the construction of U, verification of results, and interpretation of SVD in matrix decomposition.

Singular Values, Fundamental Theorem

Explores the fundamental theorem on singular values and the formation of orthonormal bases from eigenvectors.

Quadratic Forms in IR³

Explores quadratic forms in IR³, matrix properties, diagonalization, and positive definite matrices.

Classification of Quadratic Forms

Explores the classification of quadratic forms based on eigenvalues and orthogonal diagonalization of symmetric matrices.

Spectral Theorem Recap

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

Continuous Random Variables

Explores continuous random variables, density functions, joint variables, independence, and conditional densities.

Elliptical Distributions: Properties and Applications

Covers elliptical distributions, including properties, applications, and risk management implications.

Isometries and Orthonormal Bases

Discusses isometries and orthonormal bases in mathematics.

Non-Negative Definite Matrices and Covariance Matrices

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

Quadratic Forms and Symmetric Bilinear Forms

Explores quadratic forms, symmetric bilinear forms, and their properties.

Singular Value Decomposition

Covers the Singular Value Decomposition theorem and its application in decomposing matrices.

Symmetric Matrices and Eigenvectors

Covers the concept of symmetric matrices, orthogonal bases, and eigenvectors.

Hypothesis Testing with Gaussian Noise

Covers hypothesis testing with Gaussian noise and the standard Gaussian distribution.