The perimeter of an ellipse is computed numerically by an infinite series. I want to know if this series can be expressed in a simpler way. For example, via matrix exponentials. Matrix exponentials are also numerically evaluated by infinite series, so it's not obviously impossible to me. Can we find a vector $v$, and matrix $A$ such that the perimeter $P$ of an ellipse follows the formula: $$P = v^T e^A v$$
Can it be shown impossible? The matrix exponential is capable of encoding many things like geodesic paths on various manifolds so I have hope that it can express the perimeter of an ellipse as well. But it's not obvious to me how.
[Edit] $A$ and $v$ should be finite dimensional.