Matrix-order differentiation
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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[edit]
Exponential function matrix-order derivative[edit]
Let be a diagonalizable matrix. Then the matrix-order derivative of the eigenfunction of matrix order is
where can be a complex number, is the matrix whose columns consist of the eigenvectors of , and are the eigenvalues of .
Derivation of the exponential function matrix-order derivative[edit]
Observe that the n'th integer-order derivative of is given by
Allowing to be a diagonalizable matrix , we have
We evaluate (a constant raised to a matrix power) by converting it into the form of a matrix exponential
Additional reading[edit]
- 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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