# Geometric intuition for highly non-normal matrix?

From related question

1. highly non-normal linear system gives slow convergence for linear solvers. Is there geometric intuition why?

Figure in 7.2 of Corless/Fillion book :

Higham suggests using $$\tau(A)$$ the difference between squared singular value/eigenvalue magnitudes as a measure of non-normality of a matrix $$A$$:

$$\tau(A)=\sum_i |\sigma_i|^2 - |\lambda_i|^2$$

1. Is there geometric intuition for what kinds of linear operations make this measure high?

Below is a simple test in $$d=2$$, combining stretching with a rotation. $$A(\theta)=\left( \begin{array}{cc} 10 & 0 \\ 0 & 1 \\ \end{array} \right)\left( \begin{array}{cc} \cos (\theta ) & -\sin (\theta ) \\ \sin (\theta ) & \cos (\theta ) \\ \end{array} \right)$$

It seems like deviation from normality is highest $$(\tau=81)$$ when strong scaling is combined with a strong rotation $$(\theta\approx \pi/2)$$. Can this intuition be extended to higher dimensions?

Here's what eigenvalues look like for various values of $$\theta$$

Notebook