Matrix-order differentiation

Matrix-order differentiation is an extension of the derivative operator to a matrix order.

In calculus, it is standard to take derivatives of an integer order. Fractional calculus extends the order of the derivative operator to real numbers or complex numbers. In matrix order differentiation, the order of the derivative is extended to matrices.

Matrix-order derivatives

Exponential function matrix-order derivative

Let $A$ be a diagonalizable matrix. Then the matrix-order derivative of the eigenfunction $e^{kx}$ of matrix order $A$ is

$\frac{d^A}{d{x}^A}{e}^{kx}=P\left[\begin{array}{ccc}{k}^{{\lambda }_1}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & k^{{\lambda }_n}\end{array}\right]P^{-1}{e}^{kx}$

where $k$ can be a complex number, $P$ is the matrix whose columns consist of the eigenvectors of $A$, and ${\lambda }_1,...,{\lambda }_n$ are the eigenvalues of $A$.

Derivation of the exponential function matrix-order derivative

Observe that the n'th integer-order derivative of $e^{kx}$ is given by

$\frac{d^n}{d{x}^n}{e}^{kx}=k^n{e}^{kx}$

Allowing $n$ to be a diagonalizable matrix $A$, we have

$\frac{d^A}{d{x}^A}{e}^{kx}=k^A{e}^{kx}$

We evaluate $k^A$ (a constant raised to a matrix power) by converting it into the form of a matrix exponential

$\begin{array}{rl}\frac{d^A}{d{x}^A}{e}^{kx}& =e^{ln(k^A)}{e}^{kx}\\ & =e^{Aln(k)}{e}^{kx}\end{array}$

Additional reading

• Miller, G.R., & Thaheem A.B. (1997). Derivatives of matrix order. Arabian Journal of Mathematics

• Naber, M. (2003). Matrix order differintegration. Monoroe Community College, Department of Mathematics

• Porciúncula, C. (2005). Derivatives and integrals: matrix order operators as an extension of the fractional calculus. State University of Rio Grande do Sul, Brazil

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