An Approach to Matrix & Linear Algebra

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

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Section 7 : { Eigen-Blackholes } in 3-D Mapping

Topic I - Introduction & Summary for this Section :

In this Section, we demonstrate that an { eigen-blackhole } does exist in certain 3-D { linear mapping } relations.

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Topic II - Defining an { Eigen-Blackhole } :

An { eigen-blackhole } arises whenever two (2) or more { eigen-vectors } :

• share the exact-same { eigen-value } ,

• are orthogonal to one-another ,

in certian special /specific { linear mapping } relations.

Any combination of these said orthogonal { eigen-vectors } is then also an { eigen-vector } ,

• and this then forms our { eigen-blackhole } , so that :
  • every vector in the { eigen-blackhole } is an { eigen-vector } .

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Topic III - Setting up the stationary { Reference Frames } :

Let us first set up, again, the 3-Dimensional { Global Reference Frame } , as identified by the set of { orthogonal unit-vector Basis } , , & respectively ,

&

which shall remain fixed and constant throughout our analysis in this Section.

Let us also set up two (2) more { Reference Frames } :

• the { Reference Frame A } , as identified by the { orthogonal unit-vector Basis } , , & respectively , which maps exactly onto , , respectively in the { Global Reference Frame } ,

• the { Reference Frame B } , as identified by the { orthogonal unit-vector Basis } , , & respectively , which maps exactly onto , , respectively in the { Global Reference Frame } .

&

Both { Reference Frame A } & { Reference Frame B } shall also remain fixed and constant throughout our analysis in this Section.

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Topic IV - The Initial Set-up :

Let us now set up :

• a set of { orthogonal unit-vector Basis } in { Reference Frame A } , i.e. { , , } , so that this set of { orthogonal unit-vector Basis } maps exactly onto { , , } respectively, initially ,

• a set of { orthogonal unit-vector Basis } in { Reference Frame B } , i.e. { , , } , so that this set of { orthogonal unit-vector Basis } maps exactly onto { , , } respectively, initially .

as per this pair of diagrams below :

&

We also set up the intial { linear mapping } relation :

so that :

• in { Reference Frame B } maps onto , in the direction , in { Reference Frame A }
• in { Reference Frame B } maps onto , in the direction , in { Reference Frame A } , and
• in { Reference Frame B } maps onto , in the direction , in { Reference Frame A } ;

  as per this pair of diagrams below :

&

And we present also this pair of diagrams for better visualization purposes :

&

( same Set-Up as we have in { Section 4 } , initially. )

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Topic V - Creating the { Eigen-Blackhole } :

But we notice that there is { vector } on the { XQ-ZQ Plane } in { Reference Frame A } that has the exact-same length as , i.e. :

• the vector as identified in the left-hand-side diagram below :

&

We then also identify the corresponding point, { } in { Reference Frame B } on the right-hand-side diagram, so that :

• the vector in { Reference Frame B } maps onto the vector in { Reference Frame A } .

The angles & , are then as marked in the diagrams above.

Let us now take a look at this diagram below for the 2-D view :

• the { XQ-ZQ Plane } in { Reference Frame A } is on the left-hand-side, and
• the { K4-K6 Plane } in { Reference Frame B } is on the right-hand-side.

We then set up an angle , whose value is given by :

Let us now rotate the entire { Ellipse } on the left-hand-side diagram ,

• together with { } on the { Ellipse } ,

  in a counter-clockwise direction, thru this angle , to arrive at this diagram below :

We note here that the vectors & are now aligned in the same direction .

And we present this diagram below for the 3-D view :

&

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Topic VI - The Final { Linear Mapping } relation :

We then set up the final { linear mapping } relation for this Section, with :

• the { unit-vector } in { Reference Frame B } mapping onto the vector in { Reference Frame A } ,
• the { unit-vector } in { Reference Frame B } mapping onto the vector in { Reference Frame A } ,
• the { unit-vector } in { Reference Frame B } mapping onto the vector in { Reference Frame A } ;

  as per this pair of diagrams below :

&

so that :

And the final { linear mapping } relation is then :

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Topic VII - Visualizing the { Eigen-blackhole } :

We then have three (3) { eigen-vectors } for this { linear mapping } relation :

• the { unit-vector } in { Reference Frame B } mapping onto the vector in the direction in { Reference Frame A } ,
  • with an { eigen-value } equal to ;

• the { unit-vector } in { Reference Frame B } mapping onto the vector in { Reference Frame A } ,
  • also with an { eigen-value } equal to ;

• and a 3rd { eigen-vector } as marked in the diagram below :

&

( We note here that, in the special case of { = } for a particular set of { , , } ,

• the 3rd { eigen-vector } merges onto the 2nd { eigen-vector } to become a single { eigen-vector } ,

  as we have already discussed in { Sectio 6 } previously. )

We then set up this pair of diagrams below, showing the { eigen-blackhole } via the { white-color planes } :

&

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In the next Section, we shall be looking at a 4-D { eigen-blackhole } in a { 7-D Vector Space } .

go to the next Section : Section 8 - A 4-D { Eigen-blackhole } in 7-D Mapping

return to FCF's Matrix & Linear Algebra HomePage

original dated 2004-11-15