Here we look at how to decompose or factorize a Matrix into a Right Upper Triangular Matrix and an Orthogonal Matrix using Gram-Schmidt Process.

1. Gram-Schmidt and QR Decompostion
(Factorization) of Matrices
Isaac Amornortey Yowetu
NIMS-GHANA
September 24, 2020

2. 1 Introduction
2 Types of Matrices under consideration
3 QR Decompostion
4 Application of Gram-Schmidt Process to QR Decomposition
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 2 / 10

3. Introduction
Gram-Schmidt Process is one of the principal numerical
algorithms for computing QR Factorization.
Householder Triangularization is also one of the other algorithms
for computing QR Factorization.
In QR Decomposition;
Q is an Orthogonal Matrix with orthonormal basis.
R is a Right Upper Triangular Matrix.
A matrix A can be decomposed as A = QR.
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 3 / 10

4. Cases of Matrices
Considering An×m
Q-Factor
In a real space, if A is a square matrix(m=n), then
Q is an orthogonal unit vectors such that,
QT
Q = QQT
= I
if Q m × n and (m = n) with orthonormal columns, then
QT
Q = I
R-Factor
R is n × n right upper triangular matrix with rii = 0 (nonzero)
elements
R is nonsinular matrix
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 4 / 10

5. QR Decomposition
A = QR
a1 a2 · · · an = q1 q2 · · · qn





r11 r12 · · · r1n
0 r22 · · · r2n
...
...
...
...
0 0 · · · rnn





Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 5 / 10

6. Question
Example 1
Consider the matrix
B =




−1 −1 1
1 3 3
−1 −1 5
1 3 7




using Gram-Schmidt process, determine the QR Factorization.
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 6 / 10

7. solution
Rewrite B as B = [b1, b2, b3] where
b1 =




−1
1
−1
1



 , b2 =




−1
3
−1
3



 , b3 =




1
3
5
7




Suppose {v1, v2, v3} ∈ V a set of linearly independent vectors and
{q1, q2, q3} ∈ Q a set of othornormal basis.
Let v1 = b1 =




−1
1
−1
1



, r11 = ||v1|| =
√
4 = 2
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 7 / 10

8. Solution Continues...
q1 = v1
||v1||
= 1
2




−1
1
−1
1



 =




−1
2
1
2
−1
2
1
2




v2 = b2 − projq1 (b2)
v2 = b2 − r12q1
=




−1
3
−1
3



 − 4




−1
2
1
2
−1
2
1
2



 =




1
1
1
1




But, r12 = <q1,b2>
<q1,q1>
= 1
2
+ 3
2
+ 1
2
+ 3
2
= 4 and ||v2|| =
√
4 = 2
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 8 / 10

9. Solution Continues...
q2 = v2
||v2||
= 1
2




1
1
1
1



 =




1
2
1
2
1
2
1
2




v3 = b3 − projq1 (b3) − projq2 (b3)
v3 = b3 − r13q1 − r23q2
=




1
3
5
7



 − 2




−1
2
1
2
−1
2
1
2



 − 8




1
2
1
2
1
2
1
2



 =




−2
−2
2
2




But, r13 = <q1,b3>
<q1,q1>
= 2, r23 = <q2,b3>
<q2,q2>
= 8 and ||v3|| =
√
16 = 4
Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 9 / 10

10. Solution Continues...
q3 = v3
||v3||
= 1
4




−2
−2
2
2



 =




−1
2
−1
2
1
2
1
2




Solution: B = QR




−1 −1 1
1 3 3
−1 −1 5
1 3 7



 =




−1
2
1
2
−1
2
1
2
1
2
−1
2
−1
2
1
2
1
2
1
2
1
2
1
2






2 4 2
0 2 8
0 0 4


Isaac Amornortey Yowetu (NIMS-GHANA)Gram-Schmidt and QR Decompostion (Factorization) of MatricesSeptember 24, 2020 10 / 10