Derivative of matrix square root

  • Thread starter cpp6f
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  • #1
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Main Question or Discussion Point

If I have a matrix valued function A(x) of some scalar x, how do I compute the derivative of the square root of A with respect to x? It seems like it should be simple, but I can't find it anywhere on the internet. Thanks!
 

Answers and Replies

  • #2
tiny-tim
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hi cpp6f! :smile:

how would you define the square root of a matrix? :confused:

eg the square root of the 2x2 identity matrix could be itself, or [1,0;0,-1] :redface:
 
  • #3
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I forgot to mention that A is positive definitive. So the principle square root of A (the square root that is also positive definitive)
 
  • #4
lavinia
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I forgot to mention that A is positive definitive. So the principle square root of A (the square root that is also positive definitive)
See what you can do with the Chain Rule is what you are saying is A(X) is a one parameter family of positive definite square matrices.
 
  • #5
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To save you a bit of footwork - implicit differentiation on the equation S(x)*S(x)=A(x) shows that S'(x) is the unique solution to a Sylvester equation.
 

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