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Derivative of matrix square root

  1. Dec 7, 2012 #1
    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!
  2. jcsd
  3. Dec 8, 2012 #2


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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:
  4. Dec 8, 2012 #3
    I forgot to mention that A is positive definitive. So the principle square root of A (the square root that is also positive definitive)
  5. Dec 8, 2012 #4


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    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.
  6. Dec 9, 2012 #5
    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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