  Characterizing simultaneously diagonalizable (SD) matrices has been receiving
considerable attention in the recent decades due to its wide applications and
its role in matrix analysis. However, the notion of SD matrices is arguably
still restrictive for wider applications. In this paper, we consider two error
measures related to the simultaneous diagonalization of matrices, and propose
several new variants of SD thereof; in particular, TWSD, TWSD-B, T_{m,n}-SD
(SDO), DWSD and D_{m,n}-SD (SDO). Those are all weaker forms of SD. We derive
various sufficient and/or necessary conditions of them under different
assumptions, and show the relationships between these new notions. Finally, we
discuss the applications of these new notions in, e.g., quadratically
constrained quadratic programming (QCQP) and independent component analysis
(ICA).
