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Multivector (MV) functions in Clifford algebras of arbitrary dimension: Defective MV case

A. Acus, A. Dargys

Abstract

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.

Multivector (MV) functions in Clifford algebras of arbitrary dimension: Defective MV case

Abstract

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.

Paper Structure

This paper contains 9 sections, 2 theorems, 36 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and characteristic polynomial of MV
  3. Minimal polynomial of MV
  4. Calculation of MV function by classical method
  5. MV function in $\mathit{Cl}_{p,q}$ algebra
  6. Example: defective MV in $\mathit{Cl}_{3,0}$
  7. Example: defective MV in $\mathit{Cl}_{4,2}$
  8. Example: defective MV in $\mathit{Cl}_{4,2}$ with high degree irreducible polynomial
  9. Conclusions and perspective

Key Result

lemma 1

The generalized spectral decomposition of linear operator can be written as We need to show that $\mathsf{A} p_i = (\lambda_i+q_i) p_i$. Since $q_i = (\mathsf{A}-\lambda_i) p_i$, $\mathsf{A} p_i = q_i + \lambda_i p_i = q_i p_i + \lambda_i p_i$, where we used the property $p_iq_i=q_i$. Because $p_i$ is the idempontent, $p_i^2=p_i$, after expansion we have $\mathsf{A}=\mathsf

Theorems & Definitions (9)

  • lemma 1
  • proposition 1: MV function in a basis-free form
  • proof
  • remark 1: Generalization
  • remark 2: Recursion formula analogy
  • remark 3: Denominators
  • remark 4: Number of derivatives of MV function
  • remark 5: Evaluation
  • remark 6: Characteristic polynomial