Invariant functions of the elements of the powers of a square matrix

A square matrix A is raised to any real power n, negative or fractional values being permitted when Aⁿ can be defined; C is a matrix that commute with A. Linear identities existing between the elements of Aⁿ or C are investigated, in such a way that the number of elements in each identity is minimized in general. Both this number and the method of investigation depend on the Jordan canonical form of A, but if A has a special property, some of these identities and their method of derivation are independent of the Jordan canonical form.