1.2 - Matrix Operations (ERO & Echelon Forms) PDF

Preview

Summary

Download 1.2 - Matrix Operations (ERO & Echelon Forms) PDF

Description

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

1. Linear Systems For a linear equation, we represent in the following way.

𝑎1 𝑥1 + 𝑎2 𝑥2 + ⋯ + 𝑎𝑛 𝑥𝑛 = 𝑏

where 𝑎1 , 𝑎2 , ⋯ , 𝑎𝑛 and 𝑏 are real (or complex) numbers. 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 are variables. Sometimes, we name the 𝑎𝑖 as coefficient of 𝑥𝑖 and 𝑏 as constant term of the equation.

When it comes to a system of linear equation (or linear system for short), we refer to a finite numbers of linear equations in the same variables. For example, a linear system of 𝑚 equations in 𝑛 variables 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 can be written as follows 𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏2 ⋮

{𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 𝑏𝑚 Example 1: 2𝑥1 + 4𝑥2 + 0.6𝑥3 = −8 4 −3𝑥2 + 𝑥3 = 0 7 1 3.8𝑥 − 𝜋𝑥 + 𝑥 = −7 1 2 { 5 𝑛

2. Matrices of a Linear System Again, we consider a system

𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 = 𝑏1 𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 = 𝑏2 ⋮

{ 𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 = 𝑏𝑚

1

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

For brevity, the system can be rewritten as a matrix form 𝐴𝑥 = 𝑏

where i.

𝑥1 𝑥2 𝑥=( ⋮ ) 𝑥𝑛

𝑎11 𝑎21 𝐴=( ⋮ 𝑎𝑚1

(we call the matrix as coefficient matrix.)

ii.

iii.

𝑎12 𝑎22 ⋮ 𝑎𝑚2

⋯ 𝑎1𝑛 ⋯ 𝑎2𝑛 ⋱ ⋮ ) ⋯ 𝑎𝑚𝑛

𝑏1 𝑏 𝑏 = ( 2) ⋮ 𝑏𝑛

Definition 1: The Augmented Matrix of a general linear matrix can be expressed as follows 𝑎11 𝑎21 [ ⋮ 𝑎𝑚1

𝑎12 𝑎22 ⋮ 𝑎𝑚2

⋯ 𝑎1𝑛 𝑏1 ⋯ 𝑎2𝑛 𝑏2 ⋱ ⋮| ⋮] ⋯ 𝑎𝑚𝑛 𝑏𝑛

Example 2: Consider the following linear system,

𝑥1 + 𝑥2 − 2𝑥3 = 1 { 2𝑥1 − 3𝑥2 + 𝑥3 = −8 3𝑥1 + 𝑥2 + 4𝑥3 = 7

Original system 𝑥1 + 𝑥2 − 2𝑥3 = 1 { 2𝑥1 − 3𝑥2 + 𝑥3 = −8 3𝑥1 + 𝑥2 + 4𝑥3 = 7

Augmented matrix form 1 1 −2 1 [ 2 −3 1 | −8 ] 7 3 1 4

2

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

3. Elementary Row Operations (ERO) Definition 2: In general, there are 3 types of elementary row operations on any matrix:

1. 2. 3.

Multiplying all numbers/entries of a row by a (the same) non-zero constant. Summing a multiple of a row to another row, namely 𝑟𝑖 , to update the row 𝑟𝑗 . Interchanging any two different rows of the matrix.

Example 3: {

𝑥1 + 𝑥2 = 1 2𝑥1 − 3𝑥2 = −8

[

Eqn2 – 2*Eqn1

1 (− ) 𝑟2 5

(− 5)*Eqn2 1

{

𝑥1 + 𝑥2 = 1 𝑥2 = 2

[

Eqn1-Eqn2 {

𝑥1

𝑟2 − 2𝑟1

1 1 1 [ | ] 0 −5 −10

𝑥1 + 𝑥2 = 1 −5𝑥2 = −10

{

1 1 |1 ] 2 −3 −8

= −1 𝑥2 = 2

1 1 1 | ] 0 1 2 𝑟1 − 𝑟2

1 0 −1 [ ] | 0 1 2

3

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

Example 4: 1 1 −2 2 [ −3 1

𝑥1 + 𝑥2 − 2𝑥3 = 1 2𝑥 − 3𝑥2 + 𝑥3 = −8 { 1 3𝑥1 + 𝑥2 + 4𝑥3 = 7

𝑟2 − 2𝑟1 𝑟3 − 3𝑟1

Eqn2 – 2*Eqn1 Eqn3 – 3*Eqn1

1 1 −2 1 [0 −5 5 | −10] 0 −2 10 4

𝑥1 + 𝑥2 − 2𝑥3 = 1 −5𝑥2 + 5𝑥3 = −10 −2𝑥2 + 10𝑥3 = 4

{

1 (− ) 𝑟2 5

(− )*Eqn2 1

5 1

(−2) *Eqn3

{

1−8] | 7

1 (− ) 𝑟3 2

1 1 −2 1 [ 0 1 −1| 2 ] 0 1 −5 −2

𝑥1 + 𝑥2 − 2𝑥3 = 1 𝑥2 − 𝑥3 = 2 𝑥2 − 5𝑥3 = −2

𝑟3 − 𝑟2

Eqn3 – Eqn2

𝑥1 + 𝑥2 − 2𝑥3 = 1 { 𝑥2 − 𝑥3 = 2 −4𝑥3 = −4

1 1 −2 1 [ 0 1 −1| 2 ] 0 0 −4 −4 4

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

4. Row Echelon Form Before starting discussion of this section, we define the following terms.

Definition 3: i.

We call the row of a matrix to be zero row if the row consists of only zero;

ii.

We call an entry of a row to be leading entry if this entry is the first non-zero number of this row.

4.1 Row Echelon Form A matrix is named as row echelon form if it satisfies the following conditions i.

All zero row(s) is/are put near the bottom of the matrix;

ii.

The leading entry of that row is on the left hand side of the leading entry of the next row(s).

Example 5: The subsequent matrices are all in row echelon forms, i. 1 5 −3 ( ) 0 −3 −7 ii.

iii.

But, how about this one?

7 −3 28 ( 0 −7 −12) 0 0 −0.22

1 3 6 8 ( 0 0 −7 −2 ) 0 0 0 0

0 0 0 0 ) 8 (1 3 6 0 0 −7 −2 5

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

4.2 Reduced Row Echelon Form Reduced row echelon form is an extension of row echelon form if it satisfies 2 additional requirements: i.

The leading entry of each non-zero row is one;

ii.

Each leading entry 1 is the one and only one entry in its column (NOT Row!).

Example 6: The subsequent matrices are all in reduced row echelon forms, i. 1 0 −3 ( ) 0 1 −7 1 0 0 ( 0 1 0) 0 0 1

ii.

iii.

1 3 0 8 ( 0 0 1 −2 ) 0 0 0 0

Reminder: Every matrix can be reduced to a matrix in reduced row echelon form by utilizing a series of elementary row operations.

To summarize the patterns of general row echelon form and general reduced row echelon form, i.

General row echelon form a.

⊛ ∗ ∗ 0 ⊛ ∗ 0 0 0 0 0 0 0 0 0 [0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ⊛ ∗ ∗ ∗ 0 0 ⊛ ∗ ∗ 0 0 0 0 0 0 0 0 0 0 ] 6

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

b.

ii.

0 0 0 0 0 [0

⊛ 0 0 0 0 0

∗ ∗ ∗ 0 ⊛ ∗ 0 0 ⊛ ∗ 0 0 0 0 0 0 0 0 0 0 0

General reduced row echelon form a. 1 0 ∗ 0 1 ∗ 0 0 0 0 0 0 0 0 0 0 0 0 [0 0 0 0 b. 0 1 ∗ 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 [0 0 0 0

∗ ∗ ∗ ⊛ 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ⊛ ] 0

∗ 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ 1 0 ∗ ∗ 0 1 ∗ ∗ 0 0 0 0 ] 0 0 0 0

0 0 0 ∗ 0 0 ∗ 1 0 ∗ 0 1 ∗ 0 0 0 1 0 0 0 0

0 0 0 0

]

Example 6: We revisit Example 4:

𝑥1 + 𝑥2 − 2𝑥3 = 1 { 2𝑥1 − 3𝑥2 + 𝑥3 = −8 3𝑥1 + 𝑥2 + 4𝑥3 = 7

{

1 1 −2 1 [2 −3 1 | −8] 7 3 1 4

𝑟2 − 2𝑟1 𝑟3 − 3𝑟1 1 1 −2 1 [0 −5 5 | −10] 0 −2 10 4

Eqn2 – 2*Eqn1 Eqn3 – 3*Eqn1 𝑥1 + 𝑥2 − 2𝑥3 = 1 −5𝑥2 + 5𝑥3 = −10 −2𝑥2 + 10𝑥3 = 4 (−5)*Eqn2 1

(−2) *Eqn3 1

7

1 (− ) 𝑟2 5 1 (− ) 𝑟3 2

ENGG1120 2019-2020 Term 2

Topic 1.2: Matrix Operations (ERO & Echelon Forms)

𝑥33 = 𝑥1 + 𝑥𝑥22−−2𝑥 = 21 𝑥2 − 5𝑥3 = −2 Eqn3 – Eqn2

{ {

[

𝑥1 + 𝑥2 − 2𝑥3 = 1 𝑥2 − 𝑥3 = 2 −4𝑥3 = −4 1 (− 4)*Eqn3

𝑥1 + 𝑥2 { 𝑥2

=3 =3 𝑥3 = 1 Eqn1 – Eqn2

𝑥1

𝑥2

1

𝑟3 − |𝑟2

2

2

]

1 1 −2 1 [ 0 1 −1 2] | −4 0 0 1 4 (− )𝑟3 4 Achieved Row Echelon Form! 1 1 −2 1 [ 0 1 −1 | 2] 0 0 1 1 𝑟1 + 2𝑟3 𝑟3 + 𝑟2

𝑥1 + 𝑥2 − 2𝑥3 = 1 𝑥2 − 𝑥3 = 2 { 𝑥3 = 1 Eqn1 + 2*Eqn3 Eqn3 + Eqn2

{

10 1 −2 −1

1 1 03 [0 1 0| 3] 0 0 1 1 𝑟1 − 𝑟2

=0 =3 𝑥3 = 1

1 0 00 [0 1 0| 3] 0 0 1 1 Reduced Row Echelon Form!

8...