# Second order matrix differential equation in the space of symmetric positive definite matrices

In the construction of interpolations in the space of Gaussian measures, I encountered a second order matrix differential equation in the set of symmetric positive definite matrices $$\mathbb{S}_+^d\subset \mathbb{R}^{d\times d}$$ of the form $$\ddot C_t = \dot C_t C_t^{-1} \dot C_t - 2 C_t \alpha\alpha^T C_t ,$$ where $$\alpha \in \mathbb{R}^d$$ is fixed.

The system is actually a boundary value problem, with $$C_0$$ and $$C_1$$ given, however for the moment I'm happy to find any family of solutions to the equation not necessarily parametrized by the boundary values.

## Particular solutions

### $$\alpha=0$$:

For $$\alpha = 0$$ it follows that $$\frac{d}{dt} (\dot C_t C_t^{-1}) =0$$ and hence the solution is given by a matrix exponential. It can be solved for its boundary values, since the product of symmetric positive definite matrices has positive eigenvalues and hence the matrix logarithm of $$C_1C_0^{-1}$$ is well-defined and one gets $$C_t = (C_1 C_0^{-1})^t C_0 := e^{t \log C_1 C_0^{-1}} C_0$$

### $$1$$-d:

In the one dimensional situation, one has for $$\alpha \ne 0$$ the explicit solution with parameters $$\beta,\beta_0\in \mathbb{R}$$ is given by $$C_t = \frac{\beta^2}{\alpha^2 \cosh(\beta t + \beta_0)^2}$$

### Time-independent eigenbasis:

By assuming that $$C_t$$ has a fixed time-independent eigenbasis, one can generalize the $$1$$-d solution by diagonalizing the equation in this eigenbasis. However, this only encompasses a subset of all possible solutions. In particular, this only constructs solutions for boundary values $$C_0$$ and $$C_1$$ having the same eigenbasis.

## Question

Is there an explicit form of solutions to the system also for the remaining case, where the eigenbasis of $$C_t$$ might change in time?

## Further rewritings

The $$1$$-d solution suggests that $$s_t = C_t^{-1/2}$$ in $$\mathbb{S}_+^d$$ might solve an easier equation. Doing the substitution, one arrives (neglecting the explicit $$t$$-dependency) at $$\ddot s s^{-1} + s^{-1} \ddot s = \dot s s^{-2}\dot s + \dot s s^{-1} \dot s s^{-1} + s^{-1} \dot s s^{-1} \dot s - s^{-1} \dot s^2 s^{-1} + 2 s^{-1} \alpha \alpha^T s^{-1} .$$ However, this equation seems not much easier to solve, except we assume that $$\ddot s, \dot s, s$$ commute, which brings us back to the diagonal solution.

• this is a highly nonlinear set of equations; a closed-form solution is unlikely, don't you think so? Commented Jan 6, 2022 at 15:01
• You may want to take a look at Helmke & Moore's Optimization and Dynamical Systems (1996). Commented Jan 6, 2022 at 17:06
• Do you have a response to the answer on this page? Commented Jan 14, 2022 at 18:22

Mathematica cannot find the general solution of this ODE even when $$d=2$$ and $$\alpha=[1,0]^\top$$, and it cannot even find the particular solution of this ODE with the zero initial conditions. So, a closed form solution seems highly unlikely.

Here is the image of the corresponding Mathematica notebook: