An Approach to Matrix & Linear Algebra

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

Top Of Page

Section 2 : Linear Mapping Considerations

Topic I - Introduction & Summary for the Section :

In this section, we shall take a further look at the { linear mapping } relation we have previously developed in { Section 1 } .

Our key finding here is that :

• in a { linear mapping } relation, the { unit-sphere } maps onto an { ellipsoid } .

go to Top Of Page

Topic II - Recalling the { linear mapping } relation :

Let us now recall the { linear mapping } relation we have previously developed in { Section 1 } :

Let us now set up two (2) { 3-Dimensional reference frames } :

• { Reference Frame A } for the exclusive use of { X , Y , Z } , and

• { Reference Frame B } for the exclusive use of { , , } ,

  as per these two (2) diagrams below :

&

• And we present these two (2) diagrams below for better visualization purposes :

&

go to Top Of Page

Topic III - Linear Mapping Basics :

Let us now focus on the following three (3) relations of particular interests :

• the 1st relation :

  

  This then maps the { unit-vector } [ ] in { Reference Frame B } onto the [ ] vector in { Reference Frame A } , with :

  • 

• the 2nd relation :

  

  This then maps the { unit-vector } [ ] in { Reference Frame B } onto the [ ] vector in { Reference Frame A } , with :

  • 

• the 3rd relation :

  

  This then maps the { unit-vector } [ ] in { Reference Frame B } onto the [ ] vector in { Reference Frame A } , with :

  • 

as per this pair of diagrams below :

&

go to Top Of Page

Topic IV - Setting up the { Unit-Sphere } in { Reference Frame B } :

Let us now set up a { unit-sphere } in { Reference Frame B } so that :

• { } is a point on this { unit-sphere } surface , as per these two (2) diagrams below :

&

The { unit-vector } is then given by :

• with { , , } satisfying the governing equation for the { unit-sphere } , i.e. :

  

The { unit-vector } { } in { Reference Frame B } is then mapped onto the vector { } in { Reference Frame A } , as per this pair of diagrams below :

&

We shall now identify / re-label the vector { } as the vector { } , and we can write :

And the mapping relation here is then :

yielding the relations :

go to Top Of Page

Topic V - Setting up the quanity { Rho-T } :

Let us now introduce the scalar quantity { } , such that :

• [ ] is then the size of the vector [ ] .

We can then write :

yielding, on expansion :

Then, for all points on the { unit-sphere } in { Reference Frame B } , satisfying the condition :

there is a particular set of values for { , , } such that the value of is at a maximum.

go to Top Of Page

Topic VI - Identifying the Maximum :

Let us suppose that we were able to identify such a set of values for { , , } such that the value of is at a maximum.

We shall then label this set of values as the set { , , } .

• ( "HV" then stands for { associated with the Highest Value } . )

We then identify, in the diagrams below, points and vectors associated with this set , the set { , , } , which in turn is associated with the highest value of :

&

• the vector in { Reference Frame B } , as identified by the points { } & { } respectively , is then given by :

  

• this vector in { Reference Frame B } then maps onto the vector in { Reference Frame A } , as identified by the points { } & { } respectively . And is given by :

  

  where :

  • 

The scalar quantity / value , i.e. :

is then the value of associated with the set { , , } and it is the highest value for .

go to Top Of Page

Topic VII - An Analysis of the Maximum :

Let us now identify a { unit-circle } on the plane orthogonal to the { unit-vector } , in { Reference Frame B } , as per the right-hand-side diagram below :

&

This { unit-circle } in { Reference Frame B } then maps onto an { ellipse } in { Reference Frame A } , as per the left-hand-side diagram above.

Let the point, { } , be a point on the { unit-circle } in { Reference Frame B } , as per right-hand-side diagram below :

&

so that the vector in { Reference Frame B } maps onto the vector in { Reference Frame A } , as per the left-hand-side diagram above.

We then summarize the current situation as follows :

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

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

  and the { unit-vectors } & in { Reference Frame B } are orthogonal to one-another.

This then is a situation of a { 2-Dimensional Planar Mapping } of :

• a { unit-circle } mapping onto an { ellipse } ;

as per this pair of diagrams below :

&

( both the { 'new' unit-circle } & the { 'new' ellipse } are then marked in 'yellow-color' ) .

branch to Appendix B

( The reader may wish to branch to Appendix B for a briefing on { 2-Dimensional Planar Mapping } at this point . )

We then identify the angle as the angle , as per the above left-hand-side diagram above.

Let us now investigate the three (3) possibilities for the value of , as follows :

• 1st Possibility --- the angle is less than { 90 degrees } :

  

  In this case, the { unit-circle } maps onto an { ellipse } but { } is not the maximum point .

• 2nd Possibility --- the angle is exactly { 90 degrees } :

  

  In this case, the { unit-circle } maps onto an { ellipse } and { } is indeed the maximum point .

• 3rd Possibility --- the angle is more than { 90 degrees } :

  

  In this case, the { unit-circle } maps onto an { ellipse } but { } is not the maximum point .

Thus, the { 2nd Possibility } is the only viable case if { } is indeed the maximum point where attains its highest value.

That is to say :

• the angle is always 90 degrees.

go to Top Of Page

Topic VIII - The 'mapped' Ellipsoid :

The original { unit-circle } ,

• which is on the plane orthogonal the vector , in { Reference Frame B } ,

then maps onto an { ellipse } ,

• which is on the plane orthogonal the vector , in { Reference Frame A } ;