Matrix operations, The Inverse of a Matrix and Matrix Factorization

Reading: Lay, Sections 2.1, 2.2, 2.3 and 2.5 (about 24 pages).

MyMathLab: Lesson U2.1

Learning Objectives:

Basic

Compute sums, products, and scalar products of matrices.

Compute matrix transposes and products of transposed matrices.

Find the inverse of a matrix.

Use the inverse of a matrix to solve linear systems.

Factor matrices using LU factorization, and use this to solve equations.

Advanced

Demonstrate understanding of theorems and properties about matrix operations and transposes.

Prove theorems and demonstrate concept knowledge about the invertability of matrices.

Demonstrate understanding of concepts and theorems about matrix invertability.

Relate the inverse of a transformation to the inverse of its standard matrix.

Section 2.1 Matrix Operations

Video over highlights the main ideas in this section but does not provide any worked examples. This material is foundational but not particularly difficult to understand. Matrix arithmetic is exactly as you would expect it to be. However, matrix multiplication is not. Pay close attention both to its definition and two strategies for calculating it. The best way to understand matrix multiplication is through practice. The transpose of a matrix should be new, but is again rather easy to understand and compute, thought its value will not become apparent till later.

Here are some Key definitions that I used in the video.

Square Matrices

An matrix is called a square matrix

Diagonal Matrices

is diagonal, if whenever

Examples:

The Identity Matrix

where

is called the identity matrix.

Example:

The column vector of is

Matrix arithmetic

Matrix equality: Matrices and are equal if for each and .

Scalar Multiplication (Scaling): For any scalar , is a matrix with .

For

Lesson U2.1 Study GuideSunday, June 3, 2018 2:05 PM

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Matrix Addition (and subtraction) is the matrix whose entries are for all and

For

is the zero matrix, where all entries are zero. For all :

Arithmetic rules for Matrices

Matrices and scalars

Matrix multiplication

The product is defined only when the number of columns in is equal to the number of rows in , in which case and are called conformable matrices.

The product AB will have the same number of rows as A and the same number of columns of B

For

is undefined because they are not conformable

Definition

If is an matrix and if is a matrix with columns , then the product is the

matrix whose columns are:

Row-Column Rule for computing AB

In general

Warning:

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In general

In general

In general does not imply that either or

Algebraic rules for Matrix Multiplication

The Transpose of a matrix

Matrix Powers

For any square matrix , Matrix Powers has the interpretation you would expect:

Exampels: For

Section 2.2 The Inverse of Matrix

The video overview explains the key concepts of matrix inverse and includes a worked example of for matrix A. You should watch the video and then read the section in the ebook.

Here are the key definitions and theorems used in the video.

An matrix is nonsingular or invertible if there is a such that:

is the inverse of , denoted and for a given , there is only one

A singular matrix does not have an inverse

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A singular matrix does not have an inverse

If and are nonsingular then is nonsingular.

For and , if then

Finding the Inverse of a matrix:

For

If then the matrix is not invertible (singular)

If is an invertible matrix, then is invertibleIf and are invertible matrices then AB is also invertible and

If A is an invertible matrix then so is and

Theorem

Elementary Matrices

Elementary matrices are formed by performing a single elementary row operation on an identity matrix.

Type I: Interchange two rows (switch row 1 and 2)

Type II: Multiply a row by a nonzero constant (multiply row 3 by 3)

Type III: adding a multiple of 1 row to another row (add 3 times row 3 to row 1)

If is an matrix, premultiplying by , that is , performs the row operation on A that was used to generate

If is an elementary matrix, is invertible and is an elementary matrix of the same type.

Theorem

An matrix A is invertible if and only if A is row equivalent to , and in this case any sequence of row operations that reduced also transforms into

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NOTE: In the online version of the text there is an error

Start with Matrix so that the left block is and the right block is •Use elimination methods put into the row equivalent form •If then is non-singular and •Otherwise, is singular (does not have an inverse)•

Finding the Inverse of a Matrix

Example: For

find

Section 2.3

The over view video for this section is very short, as is the section.

The Invertible Matrix TheoremLet be an matrix. Then the following statements are equivalent, that is for a given A either all of the statements are true or all of the statements are false.(Note Taking Note: You should transcribe the statements from the Invertible Matrix Theorem, and as you do make sure you agree with why they must all be true.

Invertible Linear Transformations

DefinitionA linear transformation is said to be invertible if there exists a transformation

such that: for all (1)

And

for all (2)

Theorem

Let be a linear transformation and let A be the standard matrix for . Then is invertible if and only if A is inverible. In that case the linear transformation given by is the unique function satifying (1) and (2) above.

Section 2.5

There is not video overview for this section. You should read the section in the textbook. I am providing two worked examples here because I think that will be more helpful

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worked examples here because I think that will be more helpful

Example 1:

Solve the equation using the factorization provided

Example 2

Find the LU factorization of A

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