An Approach to Matrix & Linear Algebra

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

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Section 6 : Eigen-Vectors in 2-D Mapping :

Topic I - Introduction & Summary for this Section :

In this section, we shall be looking at { eigen-vectors } & { eigen-values } in 2-Dimensional Planar Mapping.

Our findings here are :

• When we map a { circle } onto a { circle } , either :
  • all vectors are { eigen-vectors } , or
  • there isn't any { eigen-vectors } , i.e. NONE .

• When we map a { circle } onto an { ellipse } , the shape and the orientation of the { Ellipse } are controlling on whether or not we have any { eigen-vectors } ,

( The reader should be familiar with the mechanics of 2-D Planar Mapping , as presented in Appendix B , before moving-on in this Section .)

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Topic II - Defining { Eigen-vectors } :

If we were able to identify, in a { linear mapping } relation, a vector such that it maps onto a vector in the same direction, then that particular vector is known as an { eiger-vector } for the { linear mapping } relation .

Let us look at a brief example here :

• Suppose our { linear mapping } relation here is :

  

  and we were able to find a set of values [ L2 , L3 ] , and a corresponding scalar value , such that :

  

  Then is the { eigenvalue } and { eigen-vector } is :

  • 

  We note here that [ -L2 , -L3 ] is also always an { eigne-vector } , i.e. :

  

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

Let us first set up the { Global Reference Frame } , identified by the orthogonal { unit-vectors } & respectively, as per this diagram below :

The { Global Reference Frame } shall then be fixed and constant throughout our analysis in this Section.

Let us also set up { Refence Frame A } & { Refernce Frame B } , so that :

• the orthogonal { unit-vectors } & in { Reference Frame A } maps exactly onto the { unit-vectors } & in the { Global Reference Frame } , respectively ,

• the orthogonal { unit-vectors } & in { Reference Frame B } maps exactly onto the { unit-vectors } & in the { Global Reference Frame } , respectively .

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Topic IV - Mapping { circles } onto { circles } :

Let us first look at a specific { linear mapping } relation where a { circle } is mapped onto a { circle } .

We then have two (2) possible cases here :

• Case One :

  In this case, we have the two (2) orthogonal { unit-vectors } & in { Reference Frame B } mapping onto the two (2) orthogonal vectors & in { Reference Frame A } , in the & directions , respectively :

  • so that the two (2) mapped vectors are the same in length,

  as per this diagram below :

  

  We then have the { unit-circle } in { Reference Frame B } mapping onto a { Circle } in { Reference Frame A }, as per this diagram below :

  

  Any point { Point W } on the { unit-circle } in { Reference Frame B } then maps onto a corresponding point { Point P } on the { Circle } in { Reference Frame A } , so that :

  • the orientation of the vector and the vector are the same ,

    because the angle is always equal to the angle when we map { circles } onto { circles } .

    ( see technical details in Case Three & Case Four below . )

  Every vector on the { unit-circle } is then an { eigen-vector } under the circumstances.

• Case Two :

  In this case, we have the two (2) orthogonal { unit-vectors } & in { Reference Frame B } mapping onto the two (2) orthogonal vectors & in { Reference Frame A } , but in orthogonal directions other than the & directions , respectively :

  • so that the two (2) mapped vectors are the same in length,

  as per this diagram below :

  

  We then also have the { unit-circle } in { Reference Frame B } mapping onto a { Circle } in { Reference Frame A }, as per this diagram below :

  

  Any point { Point W } on the { unit-circle } in { Reference Frame B } then maps onto a corresponding point { Point P } on the { Circle } in { Reference Frame A } , so that :

  • the orientation of the vector and the vector are not the same ,

    because, while the angle continues to be equal to the angle , the said orientations now differ by an angle , as per the diagram above.

    ( see technical details in Case Three & Case Four below . )

  There are then no { eigen-vectors } under the circumstances, on the { unit-circle } or otherwise, in { Reference Frame B } .

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Topic V - Mapping { circles } onto { ellipses } :

Let us now look a { linear mapping } relation, where a { unit-circle } in { Reference Frame B } is mapped onto an { ellipse } in { Reference Frame A } .

Again, there are two (2) possibilities :

• Case Three :

  In this case, we have the two (2) orthogonal { unit-vectors } & in { Reference Frame B } mapping onto the two (2) orthogonal vectors & in { Reference Frame A } , in the & directions , respectively :

  • so that the two (2) mapped vectors are not the same in length,

  as per this diagram below :

  

  We then have the { unit-circle } in { Reference Frame B } mapping onto an { Ellipse } in { Reference Frame A }, as per this diagram below :

  

  We note, first-of-all , that the vectors & are both { eigen-vectors } here , mapping onto & respectively in the respective same-directions.

  Let us now identify the lengths of the vectors & as the scalar values & respectively so that :

  • for any point { Point W } on the { unit-circle } in { Reference Frame B } mapping onto a corresponding point { Point P } on the { Ellipse } in { Reference Frame A } ,

    the vector , as identified in the diagram above, is given by :

    

    with :

    

    

  We then then have :

  

  and consequently :

  

  Let us now set up the value , for better / easier notation purposes , defined by :

  

  We can then re-write the two (2) equations above as follows :

  

  and

  

  Let us now take the { 1st derivative } of with respect to here before moving-on , using the general formula :

  

  We then have :

  

  or :

  

  Let us now define the angle as the { lag-angle } , given by :

  

  We can then write :

  

  or :

  

  yielding :

  

  consolidating terms then yields :

  

  and finally :

  

  Let us now find the maximum value for , identified as , by setting :

  

  This then yields :

  

  We then have this relation for the value of at :

  

  or :

  

  Therefore, at :

  

  Let us now bring back this equation from above :

  

  But we have established that, at :

  

  Therefore, at , we have :

  

  Finally, let us recall, from above, that :

  

  and substituting the derived values of and at , and bringing back the value of , we have :

  

  We then present this next diagram below with { as a function of } for the reader's review :

  

  And we shall find out next why is so important in determining whether { eigen-values } & { eigen-vectors } exists or not, in any given { linear mapping } relation.

• Case Four :

  In this case, we have the two (2) orthogonal { unit-vectors } & in { Reference Frame B } mapping onto the two (2) orthogonal vectors & in { Reference Frame A } , but in orthogonal directions other than the & directions , respectively :

  • so that the two (2) mapped vectors are not the same in length,

  as per this diagram below :

  

  We then have the { unit-circle } in { Reference Frame B } mapping onto an { Ellipse } in { Reference Frame A } , as per this diagram below :

  

  Again, any point { Point W } on the { unit-circle } in { Reference Frame B } then maps onto a corresponding point { Point P } on the { Ellipse } in { Reference Frame A } :

  • but the question now is whether there exists any { eigen-vectors } ?

  We note here that, if indeed we were able to find an { eigen-vector } for a particular { Point W } and the corresponding { Point P } , then the following condition must hold true :

  • 

    in order that & are in the same direction.

  Therefore, for this particular set of { Point W } & { Point P } at the { eigen-vector direction } :

  • 

  But we have already set up, previously, the { lag-angle } , i.e. :

  

  so that :

  • must now be equal to ,

    at this particular set of { Point W } & { Point P } at the { eigen-vector direction } .

  Let us now take another look at the graph for { as a function of } , as follows :

  

  We note here that :

  • For { } :

    there are four (4) points in the { 0 degrees to 360 degrees } range that satisfy the condition { = } , as marked by the 4 { red-color dots } in the above diagram ;

  • For { } or { } :

    there are exactly two (2) points, respectively, in the { 0 degrees to 360 degrees } range that satisfy the condition { = } , as marked by the 2 { dark-green-color dots } & the 2 { light-green-color dots } respectively in the above diagram ;

  • For { } :

    the condition { = } has no solution .

  For our present case here where { } , we then mark-off the two (2) pairs of { eigen-vectors } in { white-color } in this diagram below :

  

  And for the case of { } , we then mark-off the single pair of { eigen-vectors } in { white-color } in this diagram below :

  

  And for the case of { } , we then mark-off the single pair of { eigen-vectors } in { white-color } in this diagram below :

  

Let us now take a look at this next five (5) diagrams in "quick-succession" with the anlge varying from { + } to { - } :

We hope we have contributed to a fairer understanding of { eigen-vectors } & { eigen values } .

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Topic VI - Quick Re-cap :

Quick recap :

• When mapping { circles } onto { circles } , we have a { All-or-None } situation on { eigen-vectors } ,

• When mapping a { circle } onto an { ellipse } , the maximum value of the { lag-angle } , is controlling ,

  and this in turn is dependent on the ratio , i.e. :

  

  That is to say, that as moves towards { infinity } , moves towards { 90 degrees } , so that :

  • the range for , where an { eigen-vector } can exist , is now :

    { - 90 degrees } < < { + 90 degrees } ,

    i.e. the full range.

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In the next Section, we shall be looking at { eigen-vectors } in 3-D Mapping.

go to the next Section : Section 7 - The { Eigen-blackhole } in 3-D Mapping

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original dated 2004-11-15