Linear Algebra 23a: Polar Decomposition - A Product of an Orthogonal and Symmetric Matrices

[lobogato]

The enthusiasm that you show while talking is really motivating, thanks.

[pedrormbraga]

Thank you very much for such a good explanation!!!

[karsunbadminton]

Thank you very much for the wonderful video！

[joshuaronisjr]

Hi, and again, thanks for all your videos! I haven't answered your lengthy comment on the other video because the pace of my linalg class hasn't really given me time to go back and think through it (we're moving on to SVD), but I will in the next few weeks - high school is almost over, so I'll have time!


However, I have a request on this one - you're saying that although the proof is done algebraically, it has a beautiful geometric interpretation, and I accept that.

Additionally, I see how through this geometric interpretation we can prove that ALL linear transformations take circles to ellipses, and not to some other weird convoluted shapes - this is because the symmetric matrix will correspond to making an ellipse (since it scales in orthogonal directions, although perhaps not our regular x and y directions) and all the orthonormal matrix will do is rotate this ellipse. This is something I had been confused about for quite some time, and I certainly am glad to finally have an explanation for I can understand.

But, my problem is this: I can see all the beautiful geometric properties coming from the fact that it algebraically works, and then we're using those geometric properties to understand other things.

However, do you know of any resources out there, be textbooks or articles, or maybe another of your videos, that would explain this proof ENTIRELY geometrically, from the beginning, in the style of some of your other videos, or the style of 3Blue1Brown?


It's not that I don't accept the algebra - after all, the geometry is just an interpretation. Its just algebra doesn't feel intuitive, and I want to really feel like I truly understand it.

Thanks!


-Josh

[RAHUL-bdvu]

sir..you are awesome ..thanks a lot :)

[thehornandhoof]

Sir, will Q and S be unique if A is not invertible or is singular matrix, like you said at 5:45? And I am thankful for you lectures specially those you gave on Tensor Analysis.

[escher]

When something is so universal like this I start to wonder if it can be somehow generalized to non-linear transformations like diffeomorphisms

[escher]

Can we restrict the form of A using this decomposition if we know that A is nil-potent with index 2?

[dacianbonta]

@6:14 i'm the opposite.
looking at a transform as stretching a sphere into an ellipsoid is the only way it makes intuitive sense