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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?

Rodrigo de Azevedo
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MathGeek
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  • Matrix differential equations are useful for solving large systems of linear equations. Ricci calculus and tensor calculus I believe are matrix-based generalisations of calculus that I believe was developed by Einstein for his theories; generalisation in maths always allows one to solve more problems, and is never a bad thing – FShrike Aug 27 '21 at 12:57
  • As soon as you have two coupled linear differential equations you can recast the system in terms of matrices, in which case you can pull in all of linear algebra to make life easier. – DMcMor Aug 27 '21 at 14:34
  • Optimization problems in matrix variables, e.g., least-norm. – Rodrigo de Azevedo Aug 27 '21 at 15:07

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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.

user7530
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