derivative of matrix square root

cpp6f

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!

tiny-tim

hi cpp6f!

how would you define the square root of a matrix?

eg the square root of the 2x2 identity matrix could be itself, or [1,0;0,-1]

cpp6f

I forgot to mention that A is positive definitive. So the principle square root of A (the square root that is also positive definitive)

lavinia

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.

bpet

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.