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# An Approach to Matrix & Linear Algebra

#### Top Of Page

( 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

### Introduction :

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.

### Key Findings :

• 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 .