This course covers advanced topics in matrices, focusing on the Kronecker Product, Vec Operator, Spectral Decomposition, Singular Value Decomposition, Generalized Inverse Matrix, Idempotent Matrices, Projection Matrices, and more. The course teaches skills such as calculating eigenvalues, performing matrix decompositions, and applying matrix identities. The teaching method includes theoretical explanations, mathematical derivations, and practical examples. This course is intended for learners with a strong background in linear algebra and matrix theory who want to deepen their understanding of matrices and their applications in statistics and data analysis.
Overview
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.
Taught by
statisticsmatt
