1-6 Schur decomposition

Link to slides: www.dropbox.co...
In this video we show how you can
- Write a matrix in its Schur decomposition
- Use the Schur decomposition to prove that a symmetric matrix has real eigenvalues
- Use the Schur decomposition to prove that an antisymmetric matrix has purely imaginary eigenvalues
- Prove that a normal matrix has a Schur decomposition with diagonal U
- Compute the 2-norm of a matrix

[@fouziarehman4918]

In 36 (Dy,y) should be lambda1y1square+lambda2y2 square... Lambda n yn square .
Since (Dy,y) is dot product its result must be a scalar .

[@martijnanthonissen]

You are right! Thanks for letting me know!

[@artemandrienko5165]

Dear Professor, thank you very much for the explanation! How could I deal with complex matrices? Can I use QR/Schur for the complex case? As soon as I understood, you derived the explanation for the real values.

[@martijnanthonissen]

Indeed, the video is for real matrices. The decomposition exists for complex matrices too. You can look on Wikipedia to see how that works. Good luck!

[@artemandrienko5165]

Thank you very much, Professor! @@martijnanthonissen

[@jiehe6943]

proof of det is wrong at 39:36

[@martijnanthonissen]

Yes, you are right! The correct argument follows from A^T A x = lambda x, left-multiplying with A, and then setting A x equal to y. Thanks for pointing this out!