# The geometry of the transpose

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.

## Transpose properties

$(AB)^T = B^T A^T$

Proof: TODO: geometric proof

## Orthogonal matrices are rotations or reflections

Intuitively since $Q^TQ = I$ reflections and rotations

Proof:

## Symmetric matrices are orthoscaling

circular argument if I use this fact for deriving the polar decomposition?

Proof:

$A = A^T \implies QS = SQ^T \implies S = QSQ^T$

Notice $S = QSQ^T$

## The polar decomposition

You can break every transformation into scaling in orthogonal directions, then a rotation/reflection.

Proof: Let $A = QS$ for orthogonal $Q$ and symmetric/orthoscaling $S$.

Multiply by $A^T$

$A^T A = SQ^TQS = SS = S^2$

Now, if we fine $S$ we're half done, then we just need to find $Q$.

Define a "matrix square root", $\sqrt{M^2} = M$.
Compute by diagonalizing $\sqrt{M^2} = \sqrt{X \Lambda X^{-1}} = X\sqrt{\Lambda}X^{-1}$

$S^2$ is symmetric, thus we can always diagonalize. $S^2 = X\Lambda X^{-1}$