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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