I know that ordinary differentiation has many real world applications, from quantum physics to economics, but I cannot think of any real world applications of matrix differentiation. So, do any real world application exist?
2026-04-25 02:29:08.1777084148
Applications of matrix differentiation
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The simplest nontrivial functions are linear and quadratic, and these are expressed using matrices. We approximate with them, compute with them, optimize them, and do calculus with them all the time. Matrix derivatives appear in applications ranging from relativity to quantum mechanics to mechanical engineering to machine learning---I think you will be hard-pressed to find a quantitative science that doesn't use matrix derivatives in some way.
Note that matrix derivatives are no different than vector derivatives---matrices form a vector space whose basis "vectors" are just shaped differently.