An Approach to Matrix & Linear Algebra

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

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Section 8 : A 4-D { Eigen-Blackhole } in 7-D Mapping

Topic I - Introduction & Summary for this Section :

In this Section, we use a numerical procedure to demonstrate the existance of a 4-D { eigen-blackhole } in a particular 7-D { linear mapping } relation .

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Topic II - The Initial 7-D { Linear Mapping } Relation :

Let us start-off this Section with the following { Matrix RHO } for the 7-D { linear mapping } relation :

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Topic III - The Search for { equal-length vectors } :

Let us now take the number [ 13 ] and see if there are any vectors of the same length in the 7-D { ellipsoid } ,

• other than the vector of length [ 13 ] in the axial-direction.

We then identify three (3) mutually-orthogonal planes, i.e. the { TP-ZP Plane } , the { UP-YP Plane } and the { VP-XP Plane } for our search :

• For the { TP-ZP Plane } & the corresponding { K1-K7 Plane } :

  

  there is a vector of length [ 13 ] , as identified on the left-hand-side diagram.

• For the { UP-YP Plane } & the corresponding { K2-K6 Plane } :

  

  there is also a vector of length [ 13 ] , as identified on the left-hand-side diagram.

• For the { VP-XP Plane } & the corresponding { K3-K5 Plane } :

  

  there is also a vector of length [ 13 ] , as identified on the left-hand-side diagram.

We then present our findings in this summary table :

Description of the 2-D planes	Minor -Axis	Major -Axis	Value of	Value of	Value of
{ TP-ZP Plane } & { K1-K7 Plane }	5	23	32.31 degrees	71.03 degrees	38.72 degrees
{ UP-YP Plane } & { K2-K6 Plane }	7	19	38.33 degrees	65.01 degrees	26.68 degrees
{ VP-XP Plane } & { K3-K5 Plane }	11	17	32.31 degrees	44.35 degrees	12.03 degrees

with :

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Topic IV - The Final 7-D { Linear Mapping } Relation :

With the three (3) values of 's , we then use the procedure we have developed in the previous section, { Section 6 } , for rotating { planar ellipses } , to arrive at this final 7-D { linear mapping } relation :

We then have four (4) { eigen-vectors } with an { eigen-value } of [ 13 ] :

• 1st { eigen-vector } ---
• 2nd { eigen-vector } ---
• 3rd { eigen-vector } ---
• 4th { eigen-vector } ---

The other three (3) { eigen-vectors } are then :

• 5th { eigen-vector } ---
  • The { eigen-value } here is [ 8.85 ] .

• 6th { eigen-vector } ---
  • The { eigen-value } here is [ 10.23 ] .

• 7th { eigen-vector } ---
  • The { eigen-value } here is [ 14.38 ] .

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Topic V - The 4-D { Eigen-Blackhole } :

Since we now have four (4) mutually orthogonal { eigen-vectors } with the same { eigen-value } of [ 13 ] , i.e. :

• the 1st four (4) { eigen-vectors } above ,

we now do have a 4-D { Eigen-Blackhole } for this particular { Linear Mapping } relation.

The key-point about the 4-D { eigen-blackhole } here is this :

• If we are already in the { eigen-blackhole } and try to get out of the { eigen-blackhole } ,

  we can always try to move in the six (6) directions orthogonal to the particular { eigen-vector } on-hand :

  • 3 of these directions will get us out of the { eigen-blackhole } ,

  • but the other 3 will not.

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go to the next Section : Section 9 - Summary & Concluding Remarks

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original dated 2004-11-15