QR Decomposition (Gram Schmidt Method) calculator

SolutionHelp

[1-1414-21421-10] $[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]$

[1-42322] $[[1,-4],[2,3],[2,2]]$

[3-64-801] $[[3,-6],[4,-8],[0,1]]$

[124005036] $[[1,2,4],[0,0,5],[0,3,6]]$

Example

1. Find QR Decomposition (Gram Schmidt Method) ...
[1-1414-21421-10] $[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]$

Solution:

Here

A $A$

1 $1$	-1 $-1$	4 $4$
1 $1$	4 $4$	-2 $-2$
1 $1$	4 $4$	2 $2$
1 $1$	-1 $-1$	0 $0$

q1′ $q_1'$

=a1 $=a_1$

	1 $1$
	1 $1$
	1 $1$
	1 $1$

	1 $1$
	1 $1$
	1 $1$
	1 $1$

r11=||q1′||=√(1)2+(1)2+(1)2+(1)2=√4=2 $r_(11)=||q_1'||=sqrt((1)^2+(1)^2+(1)^2+(1)^2)=sqrt(4)=2$

q1=1||q1′||⋅q1′ $q_1 = 1/(||q_1'||) * q_1'$

12⋅ $1/2 *$

	1 $1$
	1 $1$
	1 $1$
	1 $1$

	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$

r12=qT1⋅a2 $r_(12)=q_1^T * a_2$

[

0.5 $0.5$

]

× $xx$

	-1 $-1$
	4 $4$
	4 $4$
	-1 $-1$

=3 $=3$

q2′ $q_2'$

=a2-r12⋅q1 $=a_2-r_(12) * q_1$

	-1 $-1$
	4 $4$
	4 $4$
	-1 $-1$

-3 $-3$

	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$

	-2.5 $-2.5$
	2.5 $2.5$
	2.5 $2.5$
	-2.5 $-2.5$

r22=||q2′||=√(-2.5)2+(2.5)2+(2.5)2+(-2.5)2=√25=5 $r_(22)=||q_2'||=sqrt((-2.5)^2+(2.5)^2+(2.5)^2+(-2.5)^2)=sqrt(25)=5$

q2=1||q2′||⋅q2′ $q_2 = 1/(||q_2'||) * q_2'$

15⋅ $1/5 *$

	-2.5 $-2.5$
	2.5 $2.5$
	2.5 $2.5$
	-2.5 $-2.5$

	-0.5 $-0.5$
	0.5 $0.5$
	0.5 $0.5$
	-0.5 $-0.5$

r13=qT1⋅a3 $r_(13)=q_1^T * a_3$

[

0.5 $0.5$

]

× $xx$

	4 $4$
	-2 $-2$
	2 $2$
	0 $0$

=2 $=2$

r23=qT2⋅a3 $r_(23)=q_2^T * a_3$

[

-0.5 $-0.5$

0.5 $0.5$

-0.5 $-0.5$

]

× $xx$

	4 $4$
	-2 $-2$
	2 $2$
	0 $0$

=-2 $=-2$

q3′ $q_3'$

=a3-r13⋅q1-r23⋅q2 $=a_3-r_(13) * q_1-r_(23) * q_2$

	4 $4$
	-2 $-2$
	2 $2$
	0 $0$

-2 $-2$

	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$
	0.5 $0.5$

+2 $+2$

	-0.5 $-0.5$
	0.5 $0.5$
	0.5 $0.5$
	-0.5 $-0.5$

	2 $2$
	-2 $-2$
	2 $2$
	-2 $-2$

r33=||q3′||=√(2)2+(-2)2+(2)2+(-2)2=√16=4 $r_(33)=||q_3'||=sqrt((2)^2+(-2)^2+(2)^2+(-2)^2)=sqrt(16)=4$

q3=1||q3′||⋅q3′ $q_3 = 1/(||q_3'||) * q_3'$

14⋅ $1/4 *$

	2 $2$
	-2 $-2$
	2 $2$
	-2 $-2$

	0.5 $0.5$
	-0.5 $-0.5$
	0.5 $0.5$
	-0.5 $-0.5$

Q $Q$

=[q1,q2,q3] $=[q_1,q_2,q_3]$

0.5 $0.5$	-0.5 $-0.5$	0.5 $0.5$
0.5 $0.5$	0.5 $0.5$	-0.5 $-0.5$
0.5 $0.5$	0.5 $0.5$	0.5 $0.5$
0.5 $0.5$	-0.5 $-0.5$	-0.5 $-0.5$

R $R$

r11 $r_(11)$	r12 $r_(12)$	r13 $r_(13)$
0 $0$	r22 $r_(22)$	r23 $r_(23)$
0 $0$	0 $0$	r33 $r_(33)$

2 $2$	3 $3$	2 $2$
0 $0$	5 $5$	-2 $-2$
0 $0$	0 $0$	4 $4$

checking

Q×R=A? $Q xx R = A?$

Q×R $Q xx R$

0.5 $0.5$	-0.5 $-0.5$	0.5 $0.5$
0.5 $0.5$	0.5 $0.5$	-0.5 $-0.5$
0.5 $0.5$	0.5 $0.5$	0.5 $0.5$
0.5 $0.5$	-0.5 $-0.5$	-0.5 $-0.5$

× $xx$

2 $2$	3 $3$	2 $2$
0 $0$	5 $5$	-2 $-2$
0 $0$	0 $0$	4 $4$

1 $1$	-1 $-1$	4 $4$
1 $1$	4 $4$	-2 $-2$
1 $1$	4 $4$	2 $2$
1 $1$	-1 $-1$	0 $0$

and

A $A$

1 $1$	-1 $-1$	4 $4$
1 $1$	4 $4$	-2 $-2$
1 $1$	4 $4$	2 $2$
1 $1$	-1 $-1$	0 $0$

Solution is possible.