Case study: linearized game dynamics

[bchasnov]

Take a 4x4 matrix split into 2x2 blocks. Parameterize each block by a diagonal component and rotation(s). For symmetric matrices, R(theta) D R(theta)^T, and for non-symmetric matrices, R(theta1) D R(theta2)^T.

Diagonals D are expressed as their average and differences (g’s and f’s).

• Knobs: four diagonal matrices with two entries, and six angles. 14 in total.
• Outputs: spectral radius, maximum eigenvalue, is stable, resolvant, quadratic numerical range, etc.

We want to especially look at the case where

1. A, D<0,
2. A \nless 0, D < 0,
3. A > 0, D < 0.

We have strong theory for the case of
M = ((A,P),(P.T,D)) and M=((A,-K),(K.T,D)),
so always be looking at those extremes.

Also, we should simulate a impulse response to the system.

Plotting into tikz will be a crucial feature.