( go to the Key Findings section below )

Section 1 | Linear Algebra for 3 Variables |

Section 2 | Linear Mapping Considerations |

Section 3 | Briefing on { othogonal unit-vector Bases } |

Section 4 | Understanding { Linear Mapping } relations |

Section 5 | Finding the { Inverse } |

Section 6 | { Eigen-vectors } in 2-D Mapping |

Section 7 | The { Eigen-Blackhole } in 3-D Mapping |

Section 8 | A 4-D { Eigen-Blackhole } in 7-D Mapping |

Section 9 | Summary & Concluding Remarks |

Appendix A | The Runge Katta Method |

Appendix B | 2-Dimensional Planar Mapping |

This paper on Matrix & Linear Algebra arose as an associated-topic in the study of the { I-CHING } , the ancient Chinese { Book of Change } , and covers work for the July-October, 2004 period.

- In a non-redundant 3-D { linear mapping } relation, a { unit-sphere } always maps onto an { ellipsoid } and this can be extended to multi-dimensions.
- The { Linear Mapping } relation is broken-down into its three (3) component parts :
- the { Matrix RHO } --- A { diagonal-elements-only } matrix which defines the { Ellipsoid } ,
- the { Matrix ALPHA } --- a { Local Orientation } matrix which allows for refinements to the { linear mapping } relation,
- the { Matrix BETA } --- a { Rotation } matrix that allows us to move the { Ellipsoid } into position for the desired { linear mapping } relation.
This allows for an interesting analysis of the { Inverse } , via { 4-Quadrant Diagrams } .

- An { Eigen-Blackhole } is based on the fact that :
- When an "up-right" { circle } maps onto an "up-right" { circle } , all vectors in the 2-D { planar mapping } relation are { eigen-vectors } .

We extend this concept to multi-dimensions for orthogonal { eigen-vectors } sharing the same { eigen-value } .

This { eigen-blackhole } can be a " dangerous" situation because :

- we can move towards an { eigen-blackhole } from one direction and exit the { eigen-blackhole } from a rather-unpredictable direction ,
- and 'all-of-a-sudden' , everthing goes wild .