Matrices

Overview

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Explore advanced matrix concepts in this comprehensive 4.5-hour video series. Delve into topics such as the Kronecker Product, Vec Operator, Spectral Decomposition, Singular Value Decomposition, and Generalized Inverse Matrices. Learn about idempotent matrices, square-root matrices, and the extended Cauchy-Schwarz Inequality. Master projection matrices, including perpendicular projection and the Gram-Schmidt Orthonormalization Process. Discover techniques for inverting partitioned matrices and working with random vectors and matrices. Understand the relationships between determinants, traces, and eigenvalues. Gain insights into the Woodbury Matrix Identity, Sherman-Morrison Formula, and the sum of perpendicular projection matrices. Enhance your understanding of matrix theory with practical applications using R programming.

Syllabus

The Kronecker Product.
The Vec Operator.
The Spectral Decomposition (Eigendecomposition).
Positive Eigenvalues of X'X and XX' are equal.
The Singular Value Decomposition.
Generalized Inverse Matrix.
Generalized Inverse for a Symmetric Matrix.
The Singular Value Decomposition (part 2).
Least Squares Inverse Matrix.
The Moore Penrose Pseudoinverse.
Idempotent Matrices.
A Square-Root Matrix.
Extended Cauchy-Schwarz Inequality.
Projection Matrices: Introduction.
Perpendicular Projection Matrix.
Gram-Schmidt Orthonormalization Process: Perpendicular Projection Matrix.
Using R: Gram-Schmidt Orthonormalization Process.
Inverse of a Partitioned Matrix.
Random Vectors and Random Matrices.
2 formulas between the determinant, trace and eigen values of a matrix.
Woodbury Matrix Identity & Sherman-Morrison Formula.
Sum of Perpendicular Projection Matrices.