An Attempt (and Failure) at Defining Fractional Calculus via Matrix Fractional Powers. Fractional calculus, traditionally defined via analytic methods such as the Riemann–Liouville and Caputo definitions, extends the concept of differentiation and integration beyond integer orders to arbitrary real orders: \[ D^\alpha f(x) = \frac{1}{\Gamma(n-\alpha)}\frac{d^n}{dx^n}\int_0^x (x-t)^{n-\alpha-1}f(t)\,dt,\quad (n=\lfloor\alpha\rfloor+1). \] Given this …