Linear Algebra 23b: How to Determine the Matrices Q and S in the Polar Decomposition

[ibrahimn]

Both the LA content and the teacher are underrated! Thank you

[mirkodobrota]

We use argument of ortogonality of Q when we were finding S square, and then we proofing ortogonality of Q with s squared? Loop?

[eamon_concannon]

Great linear algebra videos. Very useful. Thanks.

[jsluo]

No, this decomposition is not unique. It is unique iff A is invertible.

[eamon_concannon]

11.00 Did you cover the case in any video where S is not invertible?

[adarshkishore]

How would we determine the matrices Q and S in case S, and hence A isn't invertible? Btw, excellent video, thank you!

[miro.s]

You forgot to speak about the case when S is not invertible.

[kwinvdv]

If you know the eigenvalue decomposition of A^T A = P D P^T, such that S = P sqrt(D) P^T, wouldn't it be easy to find the inverse of S with S^-1 = P sqrt(D)^-1 P^T (which exists if all of the diagonal elements of D are none zero)?

[ZH]

Hi, Dr. Grinfeld. Could I ask how did you turn Q to be orthonormal matrix and get Q^T * Q = I ? Thank you.

[ZeroG]

What in the world does this have to do with "polar" anything?

If you get A^T * A = S * S, how come that does not imply A^T * A = S^T * S => A = S?