aka joint diagonalization
http://www.geo.tuwien.ac.at/downloads/tm/svd.pdf
https://math.stackexchange.com/questions/478998/a-property-of-positive-definite-matrices
A symmetric matrix and a postive-definite matrix can always be simultaneously diagonalized
Simultaneous diagnoalization can be used to solve the Generalized eigenvalue problem
See here for example, for ways of approximately simultaneously diagnoalizing Laplacian of several graphs/shapes, which can be used for several applications (in Computer graphics/Geometric deep learning). Also could be related to Grid cells... as Compatible bases on graphs
Multimodal Manifold Analysis by Simultaneous Diagonalization of Laplacians
Almost-commuting matrices are almost jointly diagonalizable – Matrix commutators: their asymptotic metric properties and relation to approximate joint diagonalization
Almost commuting self-adjoint matrices: the real and self-dual cases