Explicit formulas to calculate MV functions in a basis-free representation are presented for an arbitrary Clifford geometric algebra Cl p , q . The formulas are based on analysis of the roots of minimal MV polynomial and covers defective MVs, i.e. the MVs that have non-diagonalizable matrix representations. The method may be generalized straightforwardly to matrix functions and to finite dimensional linear operators. The results can find wide application in Clifford algebra analysis.

Formulas to calculate multivector exponentials in a basis-free representation and orthonormal basis are presented for an arbitrary Clifford geometric algebra , . The formulas are based on the analysis of roots of characteristic polynomial of a multivector. Elaborate examples how to use the formulas in practice are presented. The results are generalised to arbitrary functions of multivector and may be useful in the quantum circuits or in the problems of analysis of evolution of the entangled quantum states.

Closed form expressions in real Clifford geometric algebras Cl(0,3), Cl(3,0), Cl(1,2), and Cl(2,1) are presented in a coordinate-free form for exponential function when the exponent is a general multivector. The main difficulty in solving the problem is connected with an entanglement (or mixing) of vector and bivector components ai and ajk in a form (ai-ajk)2, i≠ j≠ k . After disentanglement, the obtained formulas simplify to the well-known Moivre-type trigonometric/hyperbolic function for vector or bivector exponentials. The presented formulas may find wide application in solving GA differential equations, in signal processing, automatic control and robotics.