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?

  • $\begingroup$ 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). $\endgroup$ – Gerry Myerson Mar 25 '19 at 9:53
  • $\begingroup$ I have updated the question now. Is it clearer in this phrasing? $\endgroup$ – LudvigH Mar 25 '19 at 10:20
  • $\begingroup$ 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. $\endgroup$ – Gerry Myerson Mar 25 '19 at 10:40
  • $\begingroup$ That is just about the type of answer I hoped for. Write it as an answer and I'll accept it. :) $\endgroup$ – 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.

  • $\begingroup$ So what do you call the three? Diagonal form, Jordan form and 'real jordan form' ? $\endgroup$ – LudvigH Mar 25 '19 at 12:43
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    $\begingroup$ The third one could be called real canonical form as at math.byu.edu/~grant/courses/m634/f99/lec11.pdf Diagonal form is just a special case of Jordan form. $\endgroup$ – Gerry Myerson Mar 25 '19 at 21:58

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