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Appl."],"published-print":{"date-parts":[[2024,3,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in 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, [Formula: see text]-SD (SDO), DWSD, and [Formula: see text]-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. 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