Matrices - Ch. 1.5, 1.9

8/4/2019 Matrices - Ch. 1.5, 1.9

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Section Two: Matrices

Textbook: Ch. 1.5, 1.9GOALS OF THIS CHAPTER

- define what a matrix is

- introduce matrix addition and properties

- introduce matrix scalar multiplication and properties

- introduce matrix transposes and properties

- understand how to write a direct proof

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION

A matrix is a rectangular array of numbers. In this course, we will usea capital letter to represent a matrix. Two smaller letters may be usedto represent the number of rows and columns:

Just what is a matrix?

“m” stands for thenumber of rows.

A mxn “n” stands for thenumber of columns.The capital “A”

stands for theentire matrix.

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION

Ex. 1 – Our first matrix

Finally, if we want to refer to one of the numbers inside the matrix,let’s say row i and column j, we write aij or a(i,j).

A 2x31 4 8

2 -7 17

=

Matrix A is “2 by 3”

I tend to use square brackets [ ] for matrices. The textbook uses roundbrackets ( ). It’s totally up to you

which you prefer!

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17

=Row 1

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17

=

Row 2

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17

=

Column 1

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17

=

Column 2

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17

=

Column 3

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION


A 2x31 4 8

2 -7 17=

a11 = a(2,3) =

The entry in the firstrow and first column

The entry in the secondrow and third column

1 17

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION

Next, we define some special types of matrices:

A row matrix is any matrix with only one row. It is of the form A 1xn.Depending on the context, a row matrix can also be called a rowvector by separating entries with commas.

A column matrix is any matrix with only one column. It is of theform Amx1. Depending on the context, a column matrix can also becalled a column vector.

A square matrix is any matrix with the same numberof rows and columns. It is of the form A nxn.

8/4/2019 Matrices - Ch. 1.5, 1.9


INTRODUCTION

Ex. 2 – Some special matrices

B 1x21 3

=

1 -1/2

3 -3/4F 2x2 =

1

24

-6

7

C 5x1 =Row Matrix

Square Matrix

Column Matrix

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION

When we add matrices together, they must be the samesize!

Ex. 3 – Matrix Addition

A 2x31 0 1

-1 2 3

= B 2x32 0 1

5 -7 1

=

A+B =

1+2

1 0 1

-1 2 3 +2 0 1

5 -7 1

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A 2x31 0 1

-1 2 3

= B 2x32 0 1

5 -7 1

=

A+B =

1+2 0+0

1 0 1

-1 2 3 +2 0 1

5 -7 1

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A 2x31 0 1

-1 2 3

= B 2x32 0 1

5 -7 1

=

A+B =

1+2 0+0 1+1

1 0 1

-1 2 3 +2 0 1

5 -7 1

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A 2x31 0 1

-1 2 3

= B 2x32 0 1

5 -7 1

=

A+B =

1+2 0+0 1+1

-1+5 2-7 3+1

1 0 1

-1 2 3 +2 0 1

5 -7 1

=

8/4/2019 Matrices - Ch. 1.5, 1.9


3 0 2

4 -5 4


A+B =

Mathematically, matrix addition looks like this:

A+B = [a ij] + [b ij] = [a ij + b ij]

(This means that we simply add corresponding entries!)

MATRIX ADDITION

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION

Ex. 4 – Failed Matrix Addition

What happens if the matrices aren’t the same size?

A 2x31 0 1

-1 2 3= B 2x2

2 0

5 -7=

A+B =

1+2

1 0 1

-1 2 3 +2 0

5 -7

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A 2x31 0 1

-1 2 3= B 2x2

2 0

5 -7=

A+B =

1+2 0+0

1 0 1

-1 2 3 +2 0

5 -7

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A 2x31 0 1

-1 2 3= B 2x2

2 0

5 -7=

A+B =

1+2 0+0 WTF

1 0 1

-1 2 3 +2 0

5 -7

=

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX ADDITION



A+B = ?

We say that the addition is not possible (or it is“undefined”) since the matrices are not the samesize.

8/4/2019 Matrices - Ch. 1.5, 1.9


PROPERTIES OF MATRIX ADDITION

Before we look at some properties, we need two

definitions.

The zero matrix is the matrix with all entries equal to zero. It isdenoted with a capital O.

The negative of a matrix A is denoted as -A. The entries of –A arethe same as A, except that all the signs are switched.

0 00 0O =

A-2 0

-5 7= -A

2 0

5 -7=

8/4/2019 Matrices - Ch. 1.5, 1.9


PROPERTIES OF MATRIX ADDITION

Appendix A: Direct Proofs- done on overhead

Thm. 5: Properties of Matrix Addition- done on overhead

Thm. 5: Proof- done on overhead

8/4/2019 Matrices - Ch. 1.5, 1.9


SCALAR MULTIPLICATION

Scalar multiplication happens when we want to multiplyall the entries in the matrix by a common number.

Ex. 6 – A Word Problem

Suppose you go Boxing Day Shopping. There are three items you arelooking for: an iPod Touch, an Xbox 360 and a Bleach DVD. At FutureShop they are having a 20% off sale and they currently have all youritems in stock! Write down a row matrix representing the price ofeach item after the discount if the iPod Touch costs $200.00, theXbox 360 costs $300.00 and the Bleach DVD costs $25.00.

8/4/2019 Matrices - Ch. 1.5, 1.9




iPodTouch

Xbox 360 BleachDVD

Price 200.00 300.00 25.00

200.00 300.00 25.00P =

8/4/2019 Matrices - Ch. 1.5, 1.9




200 300 25P – 0.2P = – 0.2 200 300 25

=200 300 25 40 60 5

–

= 160 240 20

This means the iPod Touch is $160 on sale, theXbox 360 is $240 on sale and the Bleach DVD is$20 on sale. (Taxes not included.)

8/4/2019 Matrices - Ch. 1.5, 1.9


Mathematically, scalar multiplication looks like this:


rA = r[a ij] = [ra ij]

Thm. 7: Properties of Matrix Scalar Multiplication- done on overhead

Thm. 7: Proof- done on overhead

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES

Mathematically, the matrix transpose looks like this:

At

= [a ij]t

= [a ji]

Notice the switch of the i and j! This means that therows become the columns and the columns become therows!

Howstrange…

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES

Ex. 8 – Matrix Transposes- done on overhead

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES

Some more special matrices:

A square matrix is called a diagonal matrix if there are zeros in all theentries except the main diagonal. The main diagonal entries are theentries where i=j.

A1 0

0 -3

=

Main diagonal

All other entries are zero, so A isa diagonal matrix.

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES


If all the entries of a diagonal matrix are equal to one, we call thematrix an “identity matrix.” We represent this matrix with a capital I.

I1 0

0 1

=

Main diagonal

All other entries are zero, so I is a diagonalmatrix. Since all entries on the main

diagonal are 1, it is an identity matrix.

8/4/2019 Matrices - Ch. 1.5, 1.9


8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES


A lower triangular matrix is a square matrix that has zeros in all theentries above the main diagonal. This means a(i,j)=0 if i<j.

A =

Main diagonal

All entries above the maindiagonal are zero, so A is a

lower triangular matrix.

1 0 0

7 5 0

2 -5 0

8/4/2019 Matrices - Ch. 1.5, 1.9


MATRIX TRANSPOSES


A symmetric matrix is a square matrix that satisfies A t = A. Thismeans the columns of A are also the rows of A.

A =

Main diagonal

I usually disregard the maindiagonal, then check to see if the upper triangle and lower triangle are mirror images.

1 7 2

7 5 -5

2 -5 0

8/4/2019 Matrices - Ch. 1.5, 1.9


8/4/2019 Matrices - Ch. 1.5, 1.9


PROPERTIES OF THE MATRIX TRANSPOSE

Thm. 9 – Properties of Matrix Transposes- done on overhead

Thm. 9 – Proof- done on overhead

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