Lectures related to Positive Definite Matrices

Characteristic Polynomials and Similar Matrices

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

Explores the equivalence between different properties of linear transformations represented by matrices and various matrix operations.

Linear Algebra: Matrices Properties

Explores properties of 3x3 matrices with real coefficients and determinant calculation methods.

Linear Algebra: Matrices and Operations

Introduces key concepts in linear algebra, including matrices, operations, and numerical invariants.

Matrix Multiplication: Applications and Properties

Covers matrix multiplication, properties, and inverses in linear algebra.

Matrix Equations: Finding Free Variables

Explains how to find free variables in matrix equations and analyze characteristic polynomials.

Linear Algebra Basics

Covers fundamental concepts in linear algebra, including linear equations, matrix operations, determinants, and vector spaces.

Singular Value Decomposition: Applications and Interpretation

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

Linear Algebra Review

Covers the basics of linear algebra, including matrix operations and singular value decomposition.

Characterization of Invertible Matrices

Explores the properties of invertible matrices, including unique solutions and linear independence.

Linear Algebra: Rank Theorem

Explores the Rank Theorem in linear algebra, covering matrix properties and determinants.

Convex Optimization: Linear Algebra Review

Provides a review of linear algebra concepts crucial for convex optimization, covering topics such as vector norms, eigenvalues, and positive semidefinite matrices.

Embeddability Theorem

Covers the l2-embeddability theorem, isometric embeddings, positive semidefinite matrices, and eigenvalues computation.

Bilinear Forms: Theory and Applications

Covers the theory and applications of bilinear forms in various mathematical contexts.

Matrix Operations: Linear Systems and Solutions

Explores matrix operations, linear systems, solutions, and the span of vectors in linear algebra.

Linear Systems: Diagonal and Triangular Matrices, LU Factorization

Covers linear systems, diagonal and triangular matrices, and LU factorization.

Diagonalization of Symmetric Matrices

Explores the diagonalization of symmetric matrices through orthogonal decomposition and the spectral theorem.

Linear Applications and Matrix Calculus

Explores linear applications, matrices, injectivity, unique solutions, and matrix operations.

Eigenvalues and Eigenvectors: Understanding Matrix Properties

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

Matrix Operations: Transposition, Rank, and Basis Change

Covers matrix transposition, rank, and basis change in linear algebra, exploring invertibility criteria and basis relationships.