An Approach to Matrix & Linear Algebra

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

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Section 5 : Finding the { Inverse }

Topic I - Introduction & Summary for this Section :

This Section is a continuation of { Section 4 } and we shall attempt to find the [ Inverse ] for the matrix :

In the last topic, we break the [ Inverse ] process into a [ Heart-Transplant ] operation and a [ Transpose ] operation, for further thoughts.

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Topic II - Defining the [ Inverse ] :

If we have a { Matrix R-S-T } and a { Matrix A-B-C } such that :

• where the { Matrix } on-the-left is the { Identity Matrix } ,

then, the { Matrix A-B-C } is the [ Inverse ] of the { Matrix R-S-T } .

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Topic III - Setting up the { Matrix CHI } :

Let us now set up three ( 3 ) scalar quantities / values, , , & respectively, such that :

• = { 1 / } ,
• = { 1 / } , and
• = { 1 / } .

We then set up the { Matrix CHI } as follows :

=

We note here immediately that :

and

so that the { Matrix RHO } and the { Matrix CHI } are the [ Inverse ] of one-another.

We then present this next three (3) sets of diagrams for comparison purposes, with :

• the left-hand-side diagrams based on the { Matrix RHO } , and
• the right-hand-side diagrams based on the { Matrix CHI } ;

&

&

&

for the reader's review and setting the stage for our discussion in the next [ topic ] below.

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Topic IV - Finidng the [ Inverse ] :

We then present this equation in our quest to find the [ Inverse ] :

or, in the alternate "concise" format :

arising from :

• 
• 
• 

  where is the [ Identity Matrix ] .

Based on this "long" equation above, we can then conclude that, for the { Matrix R-S-T } given by :

the [ Inverse ] , i.e. the { Matrix A-B-C } , is given by :

A rather simple derivation , "chopping" the "long" equation in halves.

We also note here that :

or, in an alternate "concise" format :

so that :

Thus, { Matrix R-S-T } is the [ Inverse ] of { Matrix A-B-C } .

• ( The proof of this may be found in many literature on { Matrix & Linear Algebra } and will be skipped-over here. )

The next question is based on this comparison for the [ SELF ] matrix , i.e. { Matrix R-S-T } , and the [ INVERSE ] matrix , i.e. { Matrix A-B-C } :

	Composition Matrices
	the Rotation Matrix	the Characteristic Matrix for the Ellipsoid	the Local Orientation Matrix
[ SELF ]
[ INVERSE ]

The intriguing question is why, for the [ INVERSE ] ,

• has now become the { Local Orientation Matrix } , and
• has now become the { Rotation Matrix } ?

A starting point here could possibly be the observation that :

• only when , and consequently ,

  can & receive the "same" treatment.

Reader's comments here ?

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Topic V - The "Heart Transplant" :

We bring in here a concept arising from the author's study of the { I-Ching } :

• the { 4-Quandrant Diagrams } .

But let us first set up two (2) more matrices, in addition to the [ SELF ] & the [ INVERSE ] :

• the [ T-O-S ] Matrix, also known as the { Transpose of the [ SELF ] } matrix, and is given by :

  [ T-O-S ] =

• the [ T-O-I ] Matrix, also known as the { Transpose of the [ INVERSE ] } matrix, and is given by :

  [ T-O-I ] =

We then put all four (4) matrices in the { 4-Quadrant Diagram } below :

[ SELF ]	[ T-O-S ]
[ T-O-I ]	[ INVERSE ]

The observation on this { 4-Quadrant Diagram } here is simply this :

• When we move { vertically } , we do a { Heart-Transplant } operation, namley :

  exchanging the { Matrix RHO } for the { Matrix CHI } , or visa-versa ;

• When we move { horizontally } , we do a { Transpose } operation .

The [ Inverse ] is then the result of two (2) operations :

• the { Heart Transplant } operation, and
• the { Transpose } operation,

  in any order / either order .

And in the { Heart Transplant } operation, each { non-zero diagonal-element } of the { Matrix RHO } is replaced with its reciprocal to form the { Matrix CHI } .

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Topic VI - The Shape of the { Ellipsoid } :

We note here that :

• the ratio { / } is always equal to the ratio { / } , so that :
  • the "boundary-condition" for both the { RHO-ellipsoid } and the { CHI-ellipsoid } are always the same ,

  • but the ratios involving the 's & 's are "reciprocating", both in value and in rank.

Let us now illustrate this point with the two (2) numerical examples below :

1st Table : ( actual values used for the ellipsoids presented in { Topic III } above )

	value of	value of	value of	ratio of /	ratio of /	ratio of /
[ Matrix RHO ]	1.800	1.200	0.900	2.000	1.500	1.333
[ Matrix CHI ]	1.111	0.833	0.556	2.000	1.333	1.500
	value of	value of	value of	ratio of /	ratio of /	ratio of /

2nd Table : ( illustrates a special feature )

	value of	value of	value of	ratio of /	ratio of /	ratio of /
[ Matrix RHO ]	1.800	1.273	0.900	2.000	1.414	1.414
[ Matrix CHI ]	1.111	0.786	0.556	2.000	1.414	1.414
	value of	value of	value of	ratio of /	ratio of /	ratio of /

We note here that :

• in the { 1st Table } , the { RHO-ellipsoid } and the { CHI-ellipsoid } have the different shapes,

• in the { 2nd Table } , the { RHO-ellipsoid } and the { CHI-ellipsoid } have the exact-same shape, because of the ratios { / } & { / } are matched exactly.

We now make these two (2) statements for the reader's further thoughts :

• 1st Statement :

  If the { diagonal-elements } of a 7 x 7 { Matrix RHO } is an { ascending geometric series } , then :

  • the { diagonal-elements } of the corresponding 7 x 7 { Matrix CHI } is a { descending geometric series } ,

    so that the shapes of the { RHO-ellipsoid } & the { CHI-ellipsoid } are the same.

• 2nd Statement :

  If the following condition for a 7 x 7 { Matrix RHO } is matched :

  • 

  then the { RHO-ellipsoid } & the { CHI-ellipsoid } have the exact-same shape.

( the ratios are matched axis-for-axis in both cases. )

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We move-on next to { eigen-vectors } & { eigen-values } , which will further demonstrate the importance of this { / } ratio.

go to the next Section : Section 6 - { Eigen-vectors } in 2-D Mapping

return to FCF's Matrix & Linear Algebra HomePage

original dated 2004-11-15