Fundamentals from Linear Algebra

Ghan S. Bhatt and Ali Sekmen

Mathematical Sciences and Computer ScienceCollege of Engineering

Tennessee State University

1st Annual Workshop on Data Sciences

Part I

Some Linear AlgebraSpectral Analysis

Singular Value Decomposition

Presenter Dr. Ghan S. Bhatt

Definitions

Range and Null Spaces

Range and Null Spaces

Definitions

Eigenvalues - Eigenvectors

Eigenvalues - Eigenvectors

Eigenvalues - Eigenvectors

Eigenvalues - Eigenvectors

Symmetric Matrices

Symmetric Matrices

Projection on a Vector

Projection on a Subspace

Singular Value Decomposition

Singular Value Decomposition

Singular Value Decomposition

Singular Value Decomposition

Important Lemma

Recall – Linear Mapping

Recall – Linear Mapping

General Matrix Norms

An Intuitive Matrix Norm

Although it is useful, it is not suitable for large set of problems and we need another definition of matrix norms

Although it is useful, it is not suitable for large set of problems and we need another definition of matrix norms

This satisfies the general matrix norm propertiesThis satisfies the general matrix norm properties

Induced Matrix Norms

Matrix p-Norm

More on Matrix Norms