QR Decomposition (Householder Method) Example [[1,-4],[2,3],[2,2]]

Find QR Decomposition (Householder Method) ...
[1-42322] $[[1,-4],[2,3],[2,2]]$

Solution:

Here

A $A$

	1 $1$	-4 $-4$
	2 $2$	3 $3$
	2 $2$	2 $2$

A1 $A_1$

	1 $1$	-4 $-4$
	2 $2$	3 $3$
	2 $2$	2 $2$

a1 $a_1$

	1 $1$
	2 $2$
	2 $2$

||a1||=√12+22+22=√9=3 $||a_1||=sqrt(1^2+2^2+2^2)=sqrt(9)=3$

v1=a1+sign(a11)||a1||e1 $v_1=a_1+sign(a_(11))||a_1||e_1$

	1 $1$
	2 $2$
	2 $2$

× $xx$

	1 $1$
	0 $0$
	0 $0$

	4 $4$
	2 $2$
	2 $2$

H1=I-2⋅v1⋅vT1vT1⋅v1 $H_1=I-2*(v_1*v_1^T)/(v_1^T*v_1)$

1	0	0
0	1	0
0	0	1

- $-$

224 $2/24$

⋅ $*$

	4 $4$
	2 $2$
	2 $2$

⋅ $*$

[

4 $4$

2 $2$

]

-13 $-1/3$	-23 $-2/3$	-23 $-2/3$
-23 $-2/3$	23 $2/3$	-13 $-1/3$
-23 $-2/3$	-13 $-1/3$	23 $2/3$

H1⋅A1 $H_1 * A_1$

-13 $-1/3$	-23 $-2/3$	-23 $-2/3$
-23 $-2/3$	23 $2/3$	-13 $-1/3$
-23 $-2/3$	-13 $-1/3$	23 $2/3$

× $xx$

	1 $1$	-4 $-4$
	2 $2$	3 $3$
	2 $2$	2 $2$

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

Now removing 1st row and 1st column, we get

A2 $A_2$

	4 $4$
	3 $3$

a2 $a_2$

	4 $4$
	3 $3$

||a2||=√42+32=√25=5 $||a_2||=sqrt(4^2+3^2)=sqrt(25)=5$

v2=a2+sign(a11)||a2||e1 $v_2=a_2+sign(a_(11))||a_2||e_1$

	4 $4$
	3 $3$

× $xx$

	1 $1$
	0 $0$

	9 $9$
	3 $3$

H2=I-2⋅v2⋅vT2vT2⋅v2 $H_2=I-2*(v_2*v_2^T)/(v_2^T*v_2)$

	1	0
	0	1

- $-$

290 $2/90$

⋅ $*$

	9 $9$
	3 $3$

⋅ $*$

[

9 $9$

3 $3$

]

	-45 $-4/5$	-35 $-3/5$
	-35 $-3/5$	45 $4/5$

H2⋅A2 $H_2 * A_2$

	-45 $-4/5$	-35 $-3/5$
	-35 $-3/5$	45 $4/5$

× $xx$

	4 $4$
	3 $3$

	-5 $-5$
	0 $0$

Since,

H2H1A=R $H_2H_1A=R$

H2H1A= $H_2H_1A=$

1 $1$	0 $0$	0 $0$
0 $0$	-45 $-4/5$	-35 $-3/5$
0 $0$	-35 $-3/5$	45 $4/5$

× $xx$

-13 $-1/3$	-23 $-2/3$	-23 $-2/3$
-23 $-2/3$	23 $2/3$	-13 $-1/3$
-23 $-2/3$	-13 $-1/3$	23 $2/3$

× $xx$

	1 $1$	-4 $-4$
	2 $2$	3 $3$
	2 $2$	2 $2$

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

= R

Also

A=H1H2R $A=H_1H_2R$ and

A=QR $A=QR$ ,

∴Q=H1H2 $:.Q=H_1H_2$

$Q=H_1H_2$ =

$-1/3$	$-2/3$	$-2/3$
$-2/3$	$2/3$	$-1/3$
$-2/3$	$-1/3$	$2/3$

$xx$

$1$	$0$	$0$
$0$	$-4/5$	$-3/5$
$0$	$-3/5$	$4/5$

$-1/3$	$14/15$	$-2/15$
$-2/3$	$-1/3$	$-2/3$
$-2/3$	$-2/15$	$11/15$

checking

$Q xx R = A?$

$Q xx R$

$-1/3$	$14/15$	$-2/15$
$-2/3$	$-1/3$	$-2/3$
$-2/3$	$-2/15$	$11/15$

$xx$

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

	$1$	$-4$
	$2$	$3$
	$2$	$2$

and

$A$

	$1$	$-4$
	$2$	$3$
	$2$	$2$

This material is intended as a summary. Use your textbook for detail explanation.
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3. Example [1-42322][[1,-4],[2,3],[2,2]]

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