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 historical and analytical foundation, a natural question arises: “Can fractional calculus be reformulated algebraically using fractional powers of linear operators?” In linear algebra, linear transformations on finite-dimensional vector spaces can be represented as square matrices, and fractional powers of these matrices can be defined rigorously using functional calculus based on the Jordan canonical form. Thus, if differentiation can be interpreted as a linear operator on an appropriate vector space, we might attempt an algebraic definition of fractional derivatives by leveraging the concept of fractional powers of matrices.
To investigate this possibility explicitly, let us consider a concrete example. Define the vector space of polynomials of degree at most three: \[ P = \{a_0 + a_1 x + a_2 x^2 + a_3 x^3 : a_i \in \mathbb{R} \text{ for all } i = 0 ,\, 1,\, 2,\, 3\}. \] Clearly, this space is four-dimensional, and we can naturally choose the standard (ordered) basis \(\{1, x, x^2, x^3\}\). With respect to this basis, the differentiation operator \(D\) is a linear transformation represented by the matrix: \[ [D] = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 \end{pmatrix}. \] We now ask a crucial question: If we attempt to define fractional calculus through fractional powers of this differentiation operator (let us call this new approach NFC: new fractional calculus), would this definition coincide with the classical fractional calculus (CFC), as defined by Riemann–Liouville and Caputo?
A careful consideration reveals a fundamental discrepancy. According to the classical (analytic) fractional calculus, applying a fractional derivative (for example, a 0.5-order derivative) to a polynomial such as \(x^3\) yields: \[ D^{0.5}(x^3) = \frac{\Gamma(4)}{\Gamma(4 - 0.5)} x^{2.5}. \] Note the resulting expression contains non-integer powers of $x$, such as $x^{2.5}$. However, within our chosen vector space $P$, the basis consists exclusively of integer powers $\{1, x, x^2, x^3\}$. Thus, any linear transformation defined algebraically within this space cannot produce fractional powers of $x$. This observation immediately highlights a fundamental incompatibility between the classical analytic definition (CFC) and the attempted algebraic definition (NFC).
This intrinsic conflict strongly suggests severe limitations of defining fractional calculus algebraically within this polynomial vector space. To gain deeper insight, I investigated whether the fractional power (specifically, the square root) of the differentiation operator \(D\) even exists at all, using the Jordan canonical form and matrix analysis.
The matrix \([D]\) defined above is nilpotent; explicitly, it is nilpotent of index 4 because \(D^4 = 0\) and \(D^3 \ne 0\). Its Jordan normal form is a single Jordan block of size 4 with eigenvalue 0. Crucially, from a linear algebraic standpoint, it is a well-known result that a nilpotent Jordan block of size greater than 1 cannot admit a square root. Through detailed analysis and explicit computations, I verified that no matrix \(S\) satisfies the equation: \[ S^2 = D. \] I carefully explored the possibility of constructing $S$ using binomial expansions and polynomial matrix functions. Yet, all attempts to find a finite-dimensional algebraic solution failed. Indeed, the existence of a square root of a nilpotent Jordan block matrix (with eigenvalue zero) is fundamentally prohibited by linear algebraic constraints.
This analytical and algebraic exploration thus yields a significant result: the fractional powers of the differentiation operator, at least within finite-dimensional polynomial spaces, simply do not exist. Consequently, defining fractional calculus purely algebraically via fractional matrix powers encounters insurmountable difficulties from the very beginning. It is clear now that NFC, as we initially envisioned, is fundamentally distinct and incompatible with CFC.
Nevertheless, this result does not categorically exclude the possibility of an algebraic fractional calculus definition in a broader context. If one were to extend the underlying vector space beyond finite-dimensional polynomial spaces—perhaps to infinite-dimensional function spaces or specialized operator spaces—an algebraic fractional calculus might indeed become viable. Such directions could serve as intriguing topics for future research.
Narin Yargui. August 1, 2025.
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