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Problem: qr decomposition house holder [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]] [ Calculator, Method and examples ]

Solution:
Your problem -> qr decomposition house holder [[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]

Here A =
 1 -1 4 1 4 -2 1 4 2 1 -1 0

A_1 =
 1 -1 4 1 4 -2 1 4 2 1 -1 0

a_1 =
 1 1 1 1

||a_1||=sqrt(1^2+1^2+1^2+1^2)=sqrt(4)=2

v_1=a_1-sign(a_(11))||a_1||e_1 =
 1 1 1 1
- 2 xx
 1 0 0 0
=
 -1 1 1 1

H_1=I - 2 *(v_1*v_1^T)/(v_1^T*v_1) =
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
-2/4*
 -1 1 1 1
*
 [ -1 1 1 1 ]
=
 1/2 1/2 1/2 1/2 1/2 1/2 -1/2 -1/2 1/2 -1/2 1/2 -1/2 1/2 -1/2 -1/2 1/2

H_1 * A_1 =
 1/2 1/2 1/2 1/2 1/2 1/2 -1/2 -1/2 1/2 -1/2 1/2 -1/2 1/2 -1/2 -1/2 1/2
xx
 1 -1 4 1 4 -2 1 4 2 1 -1 0
=
 2 3 2 0 0 0 0 0 4 0 -5 2

Now removing 1st row and 1st column, we get
A_2 =
 0 0 0 4 -5 2

a_2 =
 0 0 -5

||a_2||=sqrt(0^2+0^2+(-5)^2)=sqrt(25)=5

v_2=a_1-sign(a_(11))||a_1||e_1 =
 0 0 -5
- 5 xx
 1 0 0
=
 -5 0 -5

H_2=I - 2 *(v_1*v_1^T)/(v_1^T*v_1) =
 1 0 0 0 1 0 0 0 1
-2/50*
 -5 0 -5
*
 [ -5 0 -5 ]
=
 0 0 -1 0 1 0 -1 0 0

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