Differentiability Properties of Rotationally Invariant Functions

Let f be a function on the set Lin of all tensors (= square matrices) on a vector space of arbitrary dimension. If f is rotationally invariant (with respect to the left and right multiplication by proper orthogonal tensors), it has a representation through a symmetric even function of the signed singular values of the tensor argument A∈Lin. It is shown that f is of class C ^r ,r=0,1,...,∞, if and only if is of class C ^r , and an inductive formula is given for the derivatives D ^r f. This revised version was published online in August 2006 with corrections to the Cover Date.