How to Solve QR Decomposition Matrix - Tutorial

Definition:

QR decomposition of a matrix is otherwise known as QR factorization, which is nothing but decomposition of a matrix into an orthogonal matrix i.e product A = QR, and an upper triangular matrix R.

Example :

Consider 3 x 3 Matrix

A =	12 -51 4 6 167 -68 -4 24 -41

Solution:

Step 1:

First, split the columns of the given matrix.

V1 =

12 6 -4

V2 =

-51 167 24

V3 =

-4 68 -41

Step 2:

Let us calculate the values of R using Gram - Schmidt process.

u₁ = v₁

u₂ = v₂ - proj_u1 v₂

. . .

u_k = v_k - proj_u1 v_k - proj_u2 v_k - ... - proj_uk-1 v_k

Step 3:

proj_uj v_k = (v,u) u /(u,u)

Substitute the values in Gram - Schmidt process,

u1 =

12 6 -4

u₂ = v₂ - proj_u1 v₂

u2 =	-51 167 24	-	proju1 v2
u2 =	-51 167 24	-	(v,u) u /(u,u)
Finding (v,u) and (u,u) values
(v,u) =	-51 167 24	12 6 -4	= (-612+1002-96) = 294
(u,u) =	12 6 -4	12 6 -4	= (144+36+16) = 196
u2 =	-51 167 24	-	(294/196) 12 6 -4
u2 =	-51 167 24	-	1.5 12 6 -4
u2 =	-51 167 24	-	18 9 -6
u2 =	-69 158 30

Follow the same steps and calculate u3 matrix value,

Step 4:

U = (u1,u2,u3) =

Step 5:

Calculate the values of ||u₁||, ||u₂||, ||u₃||

Q = (u₁/||u₁||, u₂/||u₂||, u₃/||u₃||)

||u₁|| = √12² + 6² + (-4)² = √196 = 14
||u₂|| = √(-69)² + 158² + 30² = √30625 = 175
||u₃|| = √(11.6)² + 1.2² + 33² = √1225 = 35

Step 6:

Q = (u₁/||u₁||, u₂/||u₂||, u₃/||u₃||) =

Step 7:

Finally calculate the value of R

Q =



Q Transpose

Q ^T =



R = Q ^T x A
R = x



R =