This document presents definitions and examples of different types of matrices: - Orthogonal matrices satisfy AT A = I or A-1 = AT. An example 2x2 matrix H is shown to be orthogonal. - A proper orthogonal matrix has a determinant of 1. An example 2x2 matrix N is shown to be proper. - An improper orthogonal matrix has a determinant of -1. An example 2x2 matrix T is shown to be improper. - A unitary matrix has complex entries and satisfies N (N)T = (N)T N = I. An example 2x2 matrix is shown to be unitary.

1. PRESENTATION ON OF
LINEAR ALGEBRA
BY
MUHAMMAD TAHIR AZIZ (19ME140)

2. ORTHOGONAL MATRICES
•DEFINITION:
“A square matrix containing real numbers in the rows and
columns is to be orthogonal matrix if”:
AT A = I
OR
A-1 = AT

3. EXAMPLE:
• Lets consider a matrix H of order 2×2,
H=
𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥
−𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥
• Then its transpose is:
HT =
𝑐𝑜𝑠𝑥 −𝑠𝑖𝑛𝑥
𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑥
• Now we will proof the given condition i.e.
HHT =
𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑥 −𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥𝑠𝑖𝑛𝑥
−𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 + 𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑥𝑠𝑖𝑛𝑥 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑥
HHT =
1 0
0 1
• Hence H is a orthogonal matrix because it has proved the condition i.e. HT H = I
HT H = I

4. PROPER MATRIX
• DEFINITION:
“An orthogonal matrix is said to be proper if its
determinant is unity (1)”.
• Example:
Let the N be the square matrix of order 2×2:
N=
9 10
8 9
N= 81-80
N= 1
Hence N is a proper matrix.

5. IMPROPER MATRIX
• DEFINTION:
“An orthogonal matrix is said to be improper if its
determinant is negative unity (-1)”.
• EXAMPLE:
LetT be the square matrix of order 2×2:
T =
10 9
9 8
T= 80-81
T= -1
HenceT is the improper matrix.

6. UNITARY MATRIX
• DEFINITION:
An square matrix is said to be unitary matrix if it has complex
numbers and if
N (N)T = (N)T N = I
• EXAMPLE:
Let N be the square matrix of order 2×2,
N=
1
2
1 + 𝑖 1 − 𝑖
1 − 𝑖 1 + 𝑖
Then its conjugate and transpose is given by:

7. N=
1
2
1 − 𝑖 1 + 𝑖
1 + 𝑖 1 − 𝑖
Now we will prove the given condition:
N (N)T =
1
2
1 + 𝑖 1 − 𝑖
1 − 𝑖 1 + 𝑖
×
1
2
1 − 𝑖 1 + 𝑖
1 + 𝑖 1 − 𝑖
N (N)T =
1
4
4 0
0 4
N (N)T =
1 0
0 1
Hence N is a unitary matrix because it has proved the given condition.