An Approach to Matrix & Linear Algebra

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

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Section 4 : Understanding the { Linear Mapping } Relation

Topic I - Introduction & Summary for this Section :

We've found in { Section 2 } that, in any "non-redundant" 3-D { linear mapping } relation, a { unit-sphere } always maps onto an { ellipsoid } .

In this section, we start-off with an "upright" { unit-sphere } mapping onto an "upright" { ellipsoid } and move-back towards the original { linear mapping } relation.

This procedure then identifies / establishes the three (3) { component matrices } for the { linear mapping } relation :

• the { Matrix RHO } --- the { Characteristic Value Matrix } for the { ellipsoid } , and

• the two (2) { orthogonal unit-vector Basis } matrices :
  • the { Matrix ALPHA } - the { Local Orientation Matrix } , and

  • the { Matrix BETA } - the { Rotation Matrix } .

And this sets the stage for our further investigation into the { Inverse } & { eigen-vectors } in subsequent sections.

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Topic II - Setting up the { Global Reference Frame } :

Let us first set up the { Global Reference Frame } which shall remain fixed and constant throughout our analysis in this Section.

This { Global Reference Frame } is then identified by a set of three (3) orthogonal { unit-vectors } , , , and respectively, as shown below :

&

• Again, we present the diagram on-the-right for better visualization purposes.

We then also set up two (2) more { reference frames }:

• { Reference Frame A } , as identified by the set of three (3) orthogonal { unit-vectors } , , & respectively, and

• { Reference Frame B } , as identified by the set of three (3) orthogonal { unit-vectors } , , & respectively,

  as per this pair of diagrams below :

&

These two (2) { Reference Frames } shall also remain fixed and constant throughout our analysis in this Section, and :

• The { unit-vectors } , , & then maps exactly onto the { unit-vectors } , , & , respectively, in the { Global Reference Frame } ;

• The { unit-vectors } , , & then maps exactly onto the { unit-vectors } , , & , respectively, in the { Global Reference Frame } ;

  always.

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Topic III - Setting up the rotating { orthogonal unit-vector Basis } :

Let us now set up :

• a set of { orthogonal unit-vector Basis } consisting of , , & respectively, in { Reference Frame A } :
  • that maps exactly onto the { unit-vectors } , , & respectively, initially ;

• a set of { orthogonal unit-vector Basis } consisting of , , & respectively, in { Reference Frame B } :
  • that maps exactly onto the { unit-vectors } , , & respectively, initially ;

  as per this pair of diagrams below :

&

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Topic IV - Setting up the { linear mapping } relation :

Let us now pick, for our analysis purposes, three (3) scalar quantities / values, , & respectively, such that :

We then set up the { linear mapping } relation between :

• the { orthogonal unit-vector Basis - / / }

  and

• the { orthogonal unit-vector Basis - / / }

  as follows :

  

In particluar, we have the three (3) relations :

• 1st relation :

  

  This then maps the { unit-vector } in { Reference Frame B } onto the vector in the direction in { Reference Frame A } .

• 2nd relation :

  

  This then maps the { unit-vector } in { Reference Frame B } onto the vector in the direction in { Reference Frame A } .

• 3rd relation :

  

  This then maps the { unit-vector } in { Reference Frame B } onto the vector in the direction in { Reference Frame A } .

as per this pair of diagrams below :

&

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Topic V - Constructing the Ellipsoid :

We can then identify the { axial directions } for :

• the { unit-sphere } based upon / referencing the { orthogonal unit-vector Basis - / / } in { Reference Frame B } ,

  mapping onto :

• the { ellipsoid } based upon / referencing the { orthogonal unit-vector Basis - / / } in { Reference Frame A } ,

  as per this pair of diagrams below :

&

We can then construct the { unit-sphere } and the { ellipsoid } accordingly, as per this pair of diagrams below :

&

This next pair of diagrams then shows :

&

• the { K4 -K5 } plane mapping onto the { XQ - YQ } plane ,
• the { K5 -K6 } plane mapping onto the { YQ - ZQ } plane ,
• the { K6 -K4 } plane mapping onto the { ZQ - XQ } plane ,

  base upon / referencing the { orthogonal unit-vector basis - / / } and the { orthogonal unit-vector basis - / / } respectively.

  ( We will come back to this last pair of diagrams later when we discuss { eigen-vectors } in Section 6. )

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Topic VI - Selecting a { Mapping Relation } :

Let us now identify three (3) points on the { unit-sphere } , { } , { } & { } respectively :

&

such that the three (3) { unit-vectors } , & are orthogonal to one-another.

We then re-label these three (3) { unit-vectors } , & as the { unit-vectors } , , & , respectively, as per the right-hand-side diagram above.

The { unit-vectors } , & then forms a set of { orthogonal unit-vector basis } .

Let us now set up the scalar quantities , , , , , , , , & such that :

and more precisely :

The matrix above then obeys the rules associated with a [ Unit-Basis-On-Unit-Basis Mapping Matrix } , as discussed in { Section 3 } previously.

Let us now identify three (3) points on the { ellipsoid } , namely { } , { } , & { } respectively :

&

such that :

• the { unit-vectors } , , & :
  • referencing the { orthogonal unit-vector Basis - / / } in { Reference Frame B } ,

  maps onto :

• the vectors , , & :
  • referencing the { orthogonal unit-vector Basis - / / } in { Reference Frame A } ,

  respectively.

as per this set of diagrams below :

&

We then re-label the vectors , & as the vectors , & , respectively, as per this pair of diagrams below :

&

We then set up the scalar quantities , , , , , , , & such that :

yielding :

We then have the following relations :

• arising from the mapping of onto :

  

  

  

• arising from the mapping of onto :

  

  

  

• arising from the mapping of onto :

  

  

  

We then consolidate the above nine (9) equations in a single { matrix equation } , as follows :

or, in an alternately format we shall use later in the Section ( the "transposed" equation ) :

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Topic VII - Rotating the { 3-Axis } :

Let us now "move" / "rotate" the { axis of the Ellipsoid } and the { axis of the Unit-Sphere } via the exact-same set of motions :

&

so that the new orientations of the said { sets of axis } are now given by :

• 

  and

• 

  respectively,

  as per this pair of diagrams below :

&

We note that we have "moved" / "rotated" the entire { Ellipsoid } and the entire { Unit-Sphere } so that :

• the points { } , { } , & { } are now relocated to new positions, accordingly ;

• the points { } , { } , & { } are now relocated to new positions, accordingly ;

  as per the pair of diagrams below :

&

We note that the vectors , & , in { Reference Frame A } , have also been re-located, as per the left-hand-side diagram below :

&

( the right-hand-side diagram above is presented as a background reference. )

We then set up the scalar quantities , , , , , , , , & such that :

yielding :

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Topic VIII - Setting up the { Final Mapping Relation } :

Instead of the vectors { , , } , we now let :

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

And we set up this final { linear mapping } relation 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 :

&

This final { linear mapping } relation is then :

Let us now look more closely at this final { linear mapping } relation by bringing back several equations from above :

• from { Topic VI } above :

  

  and,

  

• from { Topic VII } above :

  

Combining these three (3) equations then yields :

We then compare this to an another equation from { Topic VII } immediately above, i.e. :

to arrive at :

Transposing the entire equation then yields :

This then is the most important equation, central to the concepts presented here in this Section.

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Topic IX - Re-cap & Summary :

Via this latest equation above, we have effectively broken-down the final { linear mapping } relation, i.e. :

into three (3) parts :

• the { Characteristic Value Matrix } for the Ellipsoid --- the { Matrix RHO } , or ,

• the { Local Orientation Matrix } --- the { Matrix ALPHA } , or , and its [ Transpose ] ,

• the { Rotation Matrix } , the { Matrix BETA } , or , and its [ Transpose ] .

To assist the reader to carry-thru this concept / system onto multi-dimensions, we provide herewith this table below :

Description of Vector-Space	Degrees Of Freedoms ( DOF's ) associated with			Total DOF's
	{ Matrix RHO } ( Ellipsoid )	{ Matrix ALPHA } & ( Local Orientation )	{ Matrix BETA } & ( Rotation )
3-Dimensional Vector-Space	3	3	3	9
4-Dimensional Vector-Space	4	6	6	16
5-Dimensional Vector-Space	5	10	10	25
6-Dimensional Vector-Space	6	15	15	36
7-Dimensional Vector-Space	7	21	21	49

The advantages for the concept / system presented here will be fully demonstrated in the next Section on the [ Inverse ] , and in subsequent Sections on [ eigen-vectors ] & [ eigen-values ] .

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go to the next Section : Section 5 - Finding the [ Inverse ]

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original dated 2004-11-15