Lectures related to Square matrix

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

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

Covers matrix operations, focusing on the product and inverse of matrices.

Orthogonal Matrices and Triangular Matrices

Explores properties of orthogonal and triangular matrices with linearly independent columns and their mathematical operations.

Linear Systems: Diagonal and Triangular Matrices, LU Factorization

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

Linear Applications and Eigenvalues

Covers linear applications, eigenvalues, eigenvectors, and geometric interpretations of square matrices.

Characteristic Polynomials and Similar Matrices

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

Square Matrices: Invertible, Triangular, Diagonal

Explains square matrices, including invertible, triangular, and diagonal matrices, and their properties.

Matrix Decomposition: Triangular and Spectral

Covers the decomposition of matrices into triangular blocks and spectral decomposition.

Matrix Equations: Finding Free Variables

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

Matrix Determinants: Properties and Calculations

Covers the properties and calculations of matrix determinants, including triangular matrices and elementary row operations.

Non-Negative Definite Matrices and Covariance Matrices

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

Dynamics on Homogeneous Spaces and Interactions with Number Theory

Explores dynamics on homogeneous spaces and their interactions with number theory, focusing on modular lattices and the Iwasawa decomposition theorem.

Coxeter Groups: Spectral Theorem and Sylvester's Criterion

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

Linear Systems: Matrices and Solutions

Covers linear differential systems, matrix formulation, unique solutions, and changing variables using invertible matrices.

Matrix Multiplication: Associativity Property

Explores the associativity property of matrix multiplication for proving products equal to zero.

Determinants: Triangular Matrices and Efficient Calculations

Explores efficient determinant calculations using triangular matrices and specific row/column selections for simplification.

Matrix Factorizations: LU Decomposition

Introduces LU decomposition for efficient linear equation solving using matrix factorization.

Characteristics of Matrices and Eigenvalues

Explores matrices, eigenvalues, and diagonalizability, including invertibility and vector spaces.

Eigenvalues and Diagonalization

Explores eigenvalues, eigenvectors, and matrix diagonalization with examples and proofs.