# What transparent parametrizations of 2x2 matrices are there?

I want to contruct a time series by a recurrence relation $$y_t = Ay_{t-1}$$, where A is a 2x2 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 2x2-matrices.

My question: What other parametrizations of real 2x2 matrices is there, that give instant intuition for the corresponding dynamical system?

• I don't know what you mean by parametrization of a matrix, but, if the matrix is diagonalizable, then the recurrence can be rewritten in terms of the diagonal matrix (and the eigenvectors). – Gerry Myerson Mar 25 '19 at 9:53
• I have updated the question now. Is it clearer in this phrasing? – LudvigH Mar 25 '19 at 10:20
• Every $2\times2$ real matrix is similar to a real matrix of exactly one of the following three forms: $$\pmatrix{a&0\cr0&b\cr},\pmatrix{a&1\cr0&a\cr},\pmatrix{a&-b\cr b&a}$$ Each of those gives about as much insight as there is into the behavior of the dynamical system. – Gerry Myerson Mar 25 '19 at 10:40
• That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :) – LudvigH Mar 25 '19 at 11:14

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.