Givens Rotation is one of the methods to consider in numerical analysis. It has useful application in helping to decompose a given matrix into Q and R matrices. These Q and R matrices aid in solving systems of equation. These slides clear explain how to use this Givens Rotation Method.

1. Givens Rotation Method and QR
Decomposition
Isaac Amornortey Yowetu
NIMS-GHANA
January 20, 2021

2. Introduction
Application of Givens Rotation Method to QR Decomposition
Introduction
Application of Givens Rotation Method to QR Decomposition
Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

3. Introduction
Application of Givens Rotation Method to QR Decomposition
Givens Rotation Matrix
I It can be used to decompose a matrix A into QR:
I Q: Unitary matrix
I R: Right upper triangular matrix
I It is a rotation which is said to have been spanned in a
plane by two coordinates axes.
Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

4. Introduction
Application of Givens Rotation Method to QR Decomposition
The Givens Matrix Rotation is can be represented as:
G(i, j, θ) =












1 · · · 0 · · · 0 · · · 0
.
.
.
...
.
.
.
.
.
.
.
.
.
0 · · · c · · · −s · · · 0
.
.
.
.
.
.
...
.
.
.
.
.
.
0 · · · s · · · c · · · 0
.
.
.
.
.
.
.
.
.
...
.
.
.
0 · · · 0 · · · 0 · · · 1












Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

5. Introduction
Application of Givens Rotation Method to QR Decomposition
Givens Rotation Matrix
We construct the Givens Rotation Matrix using computed c
and s from know x and y as follows
c −s
s c
x
y
=
r
0
c and s can be calculated as follows:
c = cos(θ) =
x
r
s = sin(θ) = −
y
r
r =
p
x2 + y2
Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

6. Introduction
Application of Givens Rotation Method to QR Decomposition
Application of Givens Rotation Method to QR
Decomposition
Question
Consider the matrix
A =


0 −1 1
4 2 0
3 4 0


using Givens Rotation method, determine the QR
decomposition.
Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

7. Introduction
Application of Givens Rotation Method to QR Decomposition
Solution
We consider x = 0, y = 4 and r = 4
c =
0
4
= 0 and s = −
4
4
= −1
G1 =


c −s 0
s c 0
0 0 1


G1A1 =


0 1 0
−1 0 0
0 0 1




0 −1 1
4 2 0
3 4 0

 =


4 2 0
0 −1 1
3 4 0


Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

8. Introduction
Application of Givens Rotation Method to QR Decomposition
Solution Continues...
A2 =


4 2 0
0 −1 1
3 4 0


We consider x = 4, y = 3 and r = 5
c =
4
5
= 0.8 and s = −
3
5
= −0.6
G2 =


c 0 −s
0 1 0
c 0 s


G2A2 =


0.8 0 0.6
0 1 0
−0.6 0 0.8




4 2 0
0 −1 1
3 4 0

 =


5 4 0
0 1 −1
0 2 0


Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

9. Introduction
Application of Givens Rotation Method to QR Decomposition
Solution Continues...
A3 =


5 4 0
0 1 −1
0 2 0


We consider x = 1, y = 2 and r = 2.236
c =
1
2.236
= 0.447 and s = −
2
2.236
= −0.894
G3 =


1 0 0
0 c −s
0 s c


G3A3 =


1 0 0
0 0.447 0.894
0 −0.894 0.447




5 4 0
0 1 −1
0 2 0

 =


5 4 0
0 2.236 −0.447
0 0 0.894


Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

10. Introduction
Application of Givens Rotation Method to QR Decomposition
Solution
Q = G1T
G2T
G3T
=


0 −1 0
1 0 0
0 0 1




0.8 0 −0.6
0 1 0
0.6 0 0.8




1 0 0
0 0.447 −0.894
0 0.894 0.447


Q =


0 −0.447 0.894
0.8 −0.537 −0.268
0.6 0.716 0.358


R =


5 4 0
0 2.236 −0.447
0 0 0.894


Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition

11. Introduction
Application of Givens Rotation Method to QR Decomposition
THANK YOU
Reference
https://en.wikipedia.org/wiki/Givens_rotation
Isaac Amornortey Yowetu Givens Rotation Method and QR Decomposition