Great to hear that, thank you!

Thanks! Great to hear that you like the video

You are most welcome. Thanks for your nice comment!

Thanks! Glad you like the video!

You're very welcome!

I do not have a direct answer to your question, but the video kzfaq.info/get/bejne/ha2CqLVj2rW7hYU.htmlsi=O36_V4M2V25w5vzU covers Gram-Schmidt to factor a matrix. You may also use Householder to get such a factorization

Not sure what you mean by that

Please note that you normally don’t compute the Householder matrix H. You just store the vector x that generates H

@@martijnanthonissen if i want to reduce matrix A to Hessenberg form I should multiply matrix A by Householder matrices from both sides You showed how to multiply matrix A from left and it is enough to decompose matrix A into QR moreover it makes your video the best on youtube

@@holyshit922 Thanks! Great to hear you like it!

The denominator is a scalar (a number) so the transpose is the same. Does that help?

@@martijnanthonissen yes...thank you very much

Take them or teach them?! Thanks!

Once you have found the vector x, you can compute H using H = I - 2 x x^T / x^T x. However, you usually just store the vector x. You can use the technique discussed on slide 4 to compute H y without actually computing the matrix H

@@martijnanthonissen Thanks for the quick response. Got it! I wish we had professors like you in Aachen. :) Have a great evening!

@@martijnanthonissen I still don't understand. With x1 you can find the effects of H1A (a1 having two zeros), and then with x2 we can find H2H1A (a2 having one zero). But nowhere in this is H1 and H2 is actually computed. Is it at this point, in solving Rx = H2H1b, that H1 and H2 (and thus H2H1) is computed? Thank you.

@@toanoradian Thanks for asking! The idea is that you do not compute the matrices H1 and H2. Instead you store x1 and x2, and you first compute H1*b using x1. Say c = H1*b. Next you compute H2*c using x2. Then you have found H2*c = H2*H1*b. Hope this helps!

@@martijnanthonissen 😮Of course! Just like how one can compute H1a1, one could also compute H1b! H1a1 is just a1 minus some multiples of x1 (32:11 onwards), so equally one could compute H1b by subtracting from b some multiples of x1! And as H2 is actually a matrix containing 1 in the first diagonal and a H2 hat submatrix, H2(H1b) will not change the first coordinate of (H1b), as you've shown in 43:11. I got it now! I've got it in my head that you must compute and store the Householder matrices somewhere, even if just in the way of forming Q, but you've shown that there is a method to do it without that! Again, thank you very much, professor Martijn.

There are a couple of books I like on the topic. Each one of these is a great resource: - Michael T. Heath, Scientific Computing. An introductory survey. Mc Graw Hill - Walter Gander, Martin J. Gander, Felix Kwok, Scientific Computing --- An Introduction using Maple and MATLAB. Springer, 2014 - Richard L. Burden, J. Douglas Faires and Annette M. Burden, Numerical Analysis 10th edition. Cengage Learning, 2016

Sorry about that. Some programs lower the mic volume automatically. I’ll check better before recording next time!

Yes but video is good and presentation is well done because it is shown how to multiply from the left by H matrices and it is enough for QR decomposition For Hessenberg reduction we also need to know how to multiply by H matrix from the right

Using a microphone is better.

@@martijnanthonissen You can always reupload You know better than I how to make audio louder

If you're using chrome, you can paste the following code on your address bar, replace the "javascript:" at the beginning (chrome automatically removes it) and then write e.g. 9 on the popup that comes up when you press enter, that will boost the audio of the video element that's currently playing, i.e. the video. Always check what the code does before blindly pasting code from random people on youtube comments, though. javascript:(function() { if(!window.boosterGainNode) { const video = document.querySelector('video'); const audioCtx = new AudioContext(); const mediaSource = audioCtx.createMediaElementSource(video); const gainNode = audioCtx.createGain(); mediaSource.connect(gainNode); gainNode.connect(audioCtx.destination); window.boosterGainNode = gainNode; } window.boosterGainNode.gain.value = parseFloat(prompt('Enter Boost Level. eg: 3 (enter 1 to reset)')) ?? 1; })();