Angle between the singular vectors of a matrix A and the singular vectors of Transpose(A)

Question

There is a clear relation between the eigenvectors of $A$ and $A^T$. They are mutually orthogonal.

But I cannot find a similar relation between the singular vectors of $A$ and $A^T$.

I am looking for an expression like $\cos$$($"angle between singular vectors of $A$ and $A^T$"$)=$...

Answer 1

One day later:

Referring to the articles of EngleField & Farr on eigencircles:

Englefield, M. J., & Farr, G. E. (2006). Eigencircles of 2 x 2 matrices. Mathematics Magazine, 79(4), 281 - 289

Englefield, M., & Farr, G. (2010). Eigencircles and associated surfaces. Mathematical Gazette, 94(531), 438 - 449

Let $A=$ $\begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ $f=\frac{a+d} {2}$ , $g=\frac{b-c} {2}$ ,

$\theta= -atan2(g,f) = \text{the angle between the singular vectors of }A\text{ and }A^T$

Moreover: if A is decomposed using SVD :

$ A = U \Sigma V^T$

$ U \text{ contains the singular vectors of }A$.

$ V \text{ contains the singular vectors of }A^T$.

$ u_i= \frac{A v_i}{\sigma_i} \text{ connects the singular vectors of }A\text{ and }A^T$.

$\theta$ also equals $-angle \ of \ rotation \ UV^T$ or $+ angle \ of \ rotation \ VU^T$.

Answer 2

A detailed elaboration of eigencircles for different types of 2x2 matrices can be found on:

Eigencircles of special transformations