Math V - w6, w7

Submatrix

Partitioned matrix

Partitioned matrix

Matrix Addition and Multiplication

Block matrices

can be a convenient tool to compute the inverse of a matrix or confirm the inverse does not exist:

Kronecker products

Right Kronecker product

Kronecker products

Left Kronecker product

Linear Basis Transformations

Linear Basis Transformations

Example

Transformation Theorem

Transformation Theorem

What’s P?

Transformation Theorem

Example

Singular Value Decomposition

Understanding linear basis transformations helps to understand one of the main results of this course: the singular value decomposition

Singular Value Decomposition

Following concepts to simplify discussion:

Singular Value Decomposition

The following results help us prove the singular value decomposition theorem:

Singular Value Decomposition

Singular Value Decomposition

Remark

How can we conveniently perform an SVD?

Performing SVD

Example

One-sided and generalized inverses

One sided inverses

One sided inverses

Remark

One sided inverses

We consider several useful results:

One sided inverses

Example

One-sided inverses generalize the concept of inverses we were familiar with, however

Generalized inverses

Generalized inverses

Two properties

Generalized inverses

Example

The Moore-Penrose inverse

The Moore-Penrose inverse

Properties and use

The Moore-Penrose inverse

Example