I will admit right away that there is probably not a definitive answer to this question, but I am looking for guidance, approaches, and keywords to look for.
My problem is that I have parameterized matrix $M(x)\in\mathbb{C}^{n\times n}$ where $x$ is actually $x\in\mathbb{R}^2$, but let's just consider a 1-dimensional parameter space first. Computing $M(x)$ is expensive, so I want to do so only at a sparse grid of points and interpolate for arbitrary $x$. The question is, what is the best way to do this interpolation?
Obviously, I can just, for example, linearly interpolate the matrix elements, but there are properties of the matrix that I want to preserve. For example, I know that in certain cases, the matrix is unitary, and in other cases, all I can say about it is that it is dissipative (singular values $\le1$).
How does one think about "between-ness" for matrices? Is there a spectral metric, or something in that vein? Does this get more complicated for interpolating in higher dimensional parameter spaces?
I'm aware of the interpolation of rotation matrices, where the answer is apparently related to the matrix exponential. But for cases where the matrix lacks orthogonality/unitarity, what is the right way to think about it?