Topic 1
Linear Algebra

Outline
1.
2.
3.
4.
5.
6.
7.
8.
9.

Matrix operations
Special matrices
Inverse of a square matrix
System of linear equations
Determinant of a square matrix
Partitioned matrices (block matrices)
Linear independence of vectors and dimension
Eigenvalues and Eigenvectors
Positive (negative) definite matrix

2

What is a matrix?
•

A matrix is a two dimensional rectangular array of numbers.

 a11 a12  a1n 
a
a22  a2 n 

A   21
 

 

 am1 am 2  amn 

•
•
•
•
•

We say that the dimension (or size) of is
, denoted ∈
If
, then is a square matrix
If
1, the
1 matrix is called a column ( dimensional) vector
If
1, the 1 matrix is called a row ( dimensional) vector
If
1, is just a number, called scalar

3

1. Matrix operations iff dim

dim

and

•

Equality:

•

Addition and subtraction: if dim dim + while has a typical element

•

Example: Let

for all and

2
2

2 1 3
,
1 9 0
2 9
,
14 1

•

Scalar multiplication: For a scalar ,

•

Example: Let

2
1

1
9

0
3

, then
.
1
5

2
4

has a typical element

6
, then
1
0
3
4
1

has typical element

3
, then 3
0

6
3

3
27

.

9
0

4

Transpose of a matrix:
•

For an that the ,

matrix , the transpose ′ (or ) is defined as an element of ′ is equal the , element of

 a11 a12  a1n 
 a11
a
a a22  a2 n 
21
 , then A '   12
A
 
 

 



am1 am 2  amn 

 a1n
•

Examples:

1
4

2
5

1,2,3 , then or 1,2,3
•

1
2
3

3
, then,
6

Properties of matrix transpose
–
′ ′
–
′
′
′,
′

matrix such

a21  am1  a22  am 2 


 

a2 n  amn 
4
5
6

1
2
3

′ where a is a scalar
5

Matrix