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

Proof: TODO: geometric proof

Orthogonal matrices are rotations or reflections

Intuitively since QTQ=IQ^TQ = I reflections and rotations

Proof:

Symmetric matrices are orthoscaling

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

Proof:

A=AT    QS=SQT    S=QSQTA = A^T \implies QS = SQ^T \implies S = QSQ^T

Notice S=QSQTS = QSQ^T

The polar decomposition

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

Proof: Let A=QSA = QS for orthogonal QQ and symmetric/orthoscaling SS.

Multiply by ATA^T

ATA=SQTQS=SS=S2A^T A = SQ^TQS = SS = S^2

Now, if we fine SS we're half done, then we just need to find QQ.

Define a "matrix square root", M2=M\sqrt{M^2} = M.
Compute by diagonalizing M2=XΛX1=XΛX1\sqrt{M^2} = \sqrt{X \Lambda X^{-1}} = X\sqrt{\Lambda}X^{-1}

S2S^2 is symmetric, thus we can always diagonalize. S2=XΛX1S^2 = X\Lambda X^{-1}