Consider a diagonalisable square matrix $\mathbf{A}(\theta)$ that depends on a parameter $\theta$. Suppose that the spectral decomposition for the matrix is:
$$\mathbf{A}(\theta) = \mathbf{Q} \boldsymbol{\Lambda}(\theta) \mathbf{Q}^{-1}.$$
In this decomposition, the eigenvectors do not depend on the parameter $\theta$, which seems give some useful properties. In particular, it gives $\mathbf{A}(\theta) \mathbf{A}(\theta') = \mathbf{Q} \boldsymbol{\Lambda}(\theta) \boldsymbol{\Lambda}(\theta') \mathbf{Q}^{-1}$, which makes it much simpler to deal with products of the matrix with different parameters. I am not sure if there are other useful aspects of this property, or how to look it up in the mathematical literature.
Question: Is there a name for cases like this, where the eigenvectors are invariant to the parameter of the matrix function? Does this property designate any particular class of matrix functions? What are the substantive advantages of this property for dealing with the matrix?