Householder Transformation with QR Decomposition Examples.
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- เผยแพร่เมื่อ 30 ก.ย. 2024
- Householder Reflection or Transformation is one the methods of decomposing a matrix into an Orthogonal Matrix (Q) and Right Upper Triangular Matrix (R). It helps to solve systems of equation using backward substitution.
Thank you for the explanation sir. In the first example, the signs of all the elements in the second column of Q2 is positive. The first entry of the third column of Q2 is positive and the sign of third entry is negative. i.e Q2= H1*H2=[ -0.5774 0.4083 0.7071; 0.5774 0.8165 -0.0000; -0.5774 0.4082 -0.7071]. Therefore, Q2*R2=B.
Q2= (H2 H1)^ T , Transpose is missing. Though here both are same
Thanks man. in the book it was not clear to me. you helped me a lot. thanks.
Just wanted to say your whole channel, numerical methods + examples, is absolutely solid! Thank you so much for the hard work!
Welcome. Thanks for the positive feedback.
you should mention that u1 = a1 + sign(a_11)||a1||e1. It's not always minus.
That's right. Thanks
Exactly so that H2 has - in the Submatrix where it is not and +where there is minus.
Very helpful, how come though you aren't taking the sign of first element in x to calculate v?
v_k = x+sign(x_k) ||x||_2 e_k, with k being column number on original matrix B.
In practice, both choices v=a+∥a∥e1 and v=a−∥a∥e1 work. You can try solving Question one with both signs and see the difference. I believe it will work. But if v will be zero then you need to checking the signs so mostly this is consider v=a+sign(a_11)∥a∥e1. Thanks
Super helpful, first clear and concise video explaining it I found. Thank you !!
Oh, great to know. Thanks
This is amazing help with linear algebra, I thank you so much helped a lot
Great to hear!
fucktorization lmao. love your accent
:) Thanks
Bro you are incredible. Best explanation I've heard after ages of trying to understand how to do this. Keep it up man.
Waow. Thanks for the wonderful feedback.
Great explanation, thank you very much!!
Glad you enjoyed it!
Such an amazing explanation
Good Job !!!
Glad you liked it!
Thank you for the video, I learned a lot. I would like to point out that there is a mistake on the second problem, I will not explain it here but Q = P1P2 has a calculation error. Otherwise, phenomenal video!
Yes, you are right
Thank you, great job!
Glad it was helpful!
What should we do with complex matrices?
Actually, I have not worked on any problem having complex matrix. But I believe Householder Transformation should be able to handle it. Thanks.
Thank you so much
You're most welcome
Doesn't x1 itself usually come from a matrix? Can you show or explain that?
x1 is from the matrix and and the subsequent x's are from the R's matrices. I believe the explanation is helpful.
@@isaacamornorteyyowetu how about x2?
@@nguyetmainguyenthi2292 The subsequent x's are from the R's matrices.
thanks for the discussion
Thanks for the wonderful feedback.
Thank you for the help, sir
Most welcome!
Nathan kutz covered the same topic, and only he and god knows what he is talking about.
It is nice that this video has been helpful to you. Thanks for the feedback.
In the two examples sign of first element was different but you still used - ? Is that a mistake ?
The first one was - and the second one was +. It wasn't a mistake. In practice, both choices v=a+∥a∥e1 and v=a−∥a∥e1 work. You can try solving Question one with both signs and see the difference. I believe it will work. But if v will be zero then you need to checking the signs so mostly this is consider v=a+sign(a_11)∥a∥e1. Thanks
@Haonan Yuan I think a lot needs to be studied with reference to the sign to use. But in all the examples, i maintained just one sign(-) and it worked. Kindly do a further check and help me to know on your discovery. Thanks
@@isaacamornorteyyowetu so in 2nd case it will be positive right?