QR Decomposition (Householder Method) calculator Matrix A Online Step by step

https://atozmath.com/MatrixEv.aspx?q=qrdecomphh&q1=13%2C5%2C2%2C2%3B3%2C11%2C6%2C6%3B2%2C2%2C8%2C4%3B2%2C2%2C4%2C8%60qrdecomphh%60D#tblSolution

13 5 2 2
3 11 6 6
2 2 8 4
2 2 4 8

Solution:

Here A

=

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8

A1

=

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8

a1

=

13 3 2 2

||a1||=√132+32+22+22=√186=13.6382

v1=a1-sign(a11)||a1||e1

=

13 3 2 2

-

13.6382

×

1 0 0 0

=

-0.6382 3 2 2

H1=I-2⋅v1⋅vT1vT1⋅v1

=

1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1

-

217.4073

⋅

-0.6382 3 2 2

⋅

[ -0.6382 3 2 2 ]

=

0.9532 0.22 0.1466 0.1466 0.22 -0.0341 -0.6894 -0.6894 0.1466 -0.6894 0.5404 -0.4596 0.1466 -0.6894 -0.4596 0.5404

H1⋅A1

=

0.9532 0.22 0.1466 0.1466 0.22 -0.0341 -0.6894 -0.6894 0.1466 -0.6894 0.5404 -0.4596 0.1466 -0.6894 -0.4596 0.5404

×

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8

=

13.6382 7.7723 4.986 4.986 0 -2.0322 -8.0368 -8.0368 0 -6.6881 -1.3578 -5.3578 0 -6.6881 -5.3578 -1.3578

---

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

A2

=

-2.0322 -8.0368 -8.0368 -6.6881 -1.3578 -5.3578 -6.6881 -5.3578 -1.3578

a2

=

-2.0322 -6.6881 -6.6881

||a2||=√(-2.0322)2+(-6.6881)2+(-6.6881)2=√93.5914=9.6743

v2=a1-sign(a11)||a1||e1

=

-2.0322 -6.6881 -6.6881

+

9.6743

×

1 0 0

=

-11.7064 -6.6881 -6.6881

H2=I-2⋅v1⋅vT1vT1⋅v1

=

1 0 0 0 1 0 0 0 1

-

2226.5023

⋅

-11.7064 -6.6881 -6.6881

⋅

[ -11.7064 -6.6881 -6.6881 ]

=

-0.2101 -0.6913 -0.6913 -0.6913 0.605 -0.395 -0.6913 -0.395 0.605

H2⋅A2

=

-0.2101 -0.6913 -0.6913 -0.6913 0.605 -0.395 -0.6913 -0.395 0.605

×

-2.0322 -8.0368 -8.0368 -6.6881 -1.3578 -5.3578 -6.6881 -5.3578 -1.3578

=

9.6743 6.331 6.331 0 6.8507 2.8507 0 2.8507 6.8507

---

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

A3

=

6.8507 2.8507 2.8507 6.8507

a3

=

6.8507 2.8507

||a3||=√6.85072+2.85072=√55.0588=7.4202

v3=a1-sign(a11)||a1||e1

=

6.8507 2.8507

-

7.4202

×

1 0

=

-0.5695 2.8507

H3=I-2⋅v1⋅vT1vT1⋅v1

=

1 0 0 1

-

28.4508

⋅

-0.5695 2.8507

⋅

[ -0.5695 2.8507 ]

=

0.9233 0.3842 0.3842 -0.9233

H3⋅A3

=

0.9233 0.3842 0.3842 -0.9233

×

6.8507 2.8507 2.8507 6.8507

=

7.4202 5.2639 0 -5.2298

---

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

---

Since, H3H2H1A=R

H3H2H1A=

1 0 0 0 0 1 0 0 0 0 0.9233 0.3842 0 0 0.3842 -0.9233

×

1 0 0 0 0 -0.2101 -0.6913 -0.6913 0 -0.6913 0.605 -0.395 0 -0.6913 -0.395 0.605

×

0.9532 0.22 0.1466 0.1466 0.22 -0.0341 -0.6894 -0.6894 0.1466 -0.6894 0.5404 -0.4596 0.1466 -0.6894 -0.4596 0.5404

×

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8

=

13.6382 7.7723 4.986 4.986 0 9.6743 6.331 6.331 0 0 7.4202 5.2639 0 0 0 -5.2298

= R

Also A=H1H2H3R and A=QR, ∴Q=H1H2H3

Q=H1H2H3=

0.9532 0.22 0.1466 0.1466 0.22 -0.0341 -0.6894 -0.6894 0.1466 -0.6894 0.5404 -0.4596 0.1466 -0.6894 -0.4596 0.5404

×

1 0 0 0 0 -0.2101 -0.6913 -0.6913 0 -0.6913 0.605 -0.395 0 -0.6913 -0.395 0.605

×

1 0 0 0 0 1 0 0 0 0 0.9233 0.3842 0 0 0.3842 -0.9233

=

0.9532 -0.249 -0.1586 0.0654 0.22 0.9603 -0.1586 0.0654 0.1466 0.0889 0.9037 0.3922 0.1466 0.0889 0.3647 -0.9152

---

checking Q×R=A?

Q×R

=

0.9532 -0.249 -0.1586 0.0654 0.22 0.9603 -0.1586 0.0654 0.1466 0.0889 0.9037 0.3922 0.1466 0.0889 0.3647 -0.9152

×

13.6382 7.7723 4.986 4.986 0 9.6743 6.331 6.331 0 0 7.4202 5.2639 0 0 0 -5.2298

=

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8

and A

=

13 5 2 2 3 11 6 6 2 2 8 4 2 2 4 8