An Approach to Matrix & Linear Algebra

by Frank C. Fung ( 1st published in November, 2004. )

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( go to the Key Findings section below )

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 .

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go to the 1st section : Section 1 - Linear Algebra for 3 Variables

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Original dated 2004-11-15 *** Updated 2008-11-01 / 2009-10-26