Linear Algebra 23g: SVD Proof That the Orthogonal Matrices in Alternative Polar Decompositions Match

[JulianPanetta]

This proves that there exists a pair of right and left polar decompositions (A = Q1S1, A = S2Q2) with Q1 = Q2. But, it's slightly misleading to say "Q1 = Q2" because there are additional decompositions when A is singular: A = S2(Q2Q3), where Q3 acts like the identity on A's row space but performs an arbitrary orthogonal transformation within A's nullspace.

Of course, when A is nonsingular, unique (S1, S2) are invertible so Q1 = Q2 is unique. Incidentally, a slightly more direct proof that left/right polar decompositions come in pairs with same Q:
A = Q S = Q S Q^T Q = S' Q, where S' = Q S Q^T, and likewise:
A = S Q = Q Q^T S Q = Q S', where S' = Q^T S Q.
(i.e. any given right polar decomposition can be converted into a left one, and vice versa).

Thanks for posting your lecture videos! I'm also currently enjoying your tensor calculus book.

[NaturesFirstGreen]

Thank you for these videos. You've given me a better understanding and appreciation of linear algebra. I've come to the end of this playlist but I hope there will be more!

[tangolasher]

+MathTheBeautiful At 1:57 how do you know that X * Lambda * X transpose = S1, and the other two terms = Q1? Could you please explain the reasoning for that substitution?