I want to contruct a time series by a recurrence relation $y_t = Ay_{t-1}$, where A is a 2×2 real matrix. Assume $y_0$ is some known initial condition. Also, I assume $y_i \in \mathbb R^2$.
To understand the system without having to solve the equation, it would be nice to have a parametrization of the matrix that is transparent with the behavior of the solution.
The most simple parametrization would be by the real entries in the matrix. Use $(a,b,c,d)$. Unfortunately, it is not easy to understand the dynamics of the system easily by this set of parameters.
$$ A_{a,b,c,d} = \begin{bmatrix}
a& b\\
c & d
\end{bmatrix}$$
Option two is if the matrix is (real) diagonalizable. Taking the eigenvalues $(\lambda_1,\lambda_2)$. On the downside, this will only give system dynamics that are exponential growth of decline.
$$ B_{\lambda_1,\lambda_2} = \begin{bmatrix}
\lambda_1& 0\\
0 & \lambda_2
\end{bmatrix}$$
Option three is to think of the solutions as damped pendulum solutions. Then the paramterization is $(\zeta, \omega)$, giving below matrix. The notation is inspired by https://en.wikipedia.org/wiki/Harmonic_oscillator, but it is not perfectly equivalent. In this notation, I can easily read off the dampening of the solutions, since $\zeta$ is the real part of the eigenvalues, and the frequency, since $\omega$ gives the imaginary part of the eigenvalues.
$$ C_{\zeta,\omega} = \begin{bmatrix}
\zeta& -\omega\\
\omega & \zeta
\end{bmatrix}
$$
The three examples are not satisfactory to me, since they produce either (A) solutions that are not easily coupled to the parameters, (B) noninteresting dynamics, or (C) only spans a 2-dimensional subspace of all 2×2-matrices.
My question: What other parametrizations of real 2×2 matrices is there, that give instant intuition for the corresponding dynamical system?
Best Answer
Every $2\times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$\pmatrix{a&0\cr0&b\cr},\quad\pmatrix{a&1\cr0&a\cr},\quad\pmatrix{a&-b\cr b&a\cr}$$ Each of those gives you about as much insight into the behavior of a dynamical system as you can get.