NOTE: Work in progress
Here's the plan, I'm going to assume you've watched essense of linear algebra and understand it. My goal is to deepen that visual understanding to the transpose.
Proof: TODO: geometric proof
Intuitively since reflections and rotations
circular argument if I use this fact for deriving the polar decomposition?
You can break every transformation into scaling in orthogonal directions, then a rotation/reflection.
Proof: Let for orthogonal and symmetric/orthoscaling .
Now, if we fine we're half done, then we just need to find .
Define a "matrix square root", .
Compute by diagonalizing
is symmetric, thus we can always diagonalize.