''There is a lot more to matrix analysis than this - see for example SingularValueDecomposition.''

and also http://en.wikipedia.org/wiki/Matrix_theory
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If the EigenValue's magnitude is greater than 1, 
it equals (1 + interest rate) [''or 1 + growth rate if that's clearer to you''].

If the EigenValue's magnitude is less than 1, 
it equals (1 - decay rate).

If the EigenValue's magnitude is 1, the system neither grows nor decays.
It might flip-flop or cycle, though.

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If the EigenValue is a complex number,
it indicates a cycling system.

''How about combinations of EigenValue''''''s? It's not all scaling; what makes multiplication by one matrix do a reflection, and another a rotation?''

If the EigenValue is -1,
multiplying by it does a reflection 
(which is also a 180 degree rotation).

If the EigenValue is a negative real number,
multiplying by it does a reflection (and a scaling).

If the EigenValue is i,
multiplying by it is the same as rotating 90 degrees.
After 2 rotations, you will do a reflection.
After 4 rotations, 
you will wrap all the way around.
This is a cycle.

If the EigenValue is a complex number,
multiplying by it does a rotation (and a scaling).
After several rotations, 
you will eventually wrap all the way around.
This is a cycle.
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An online Matrix Calculator is at http://wims.unice.fr/wims/wims.cgi?session=3G0DBDBD76.5&+lang=en&+module=tool%2Flinear%2Fmatrix.en
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See also 

* ComplexNumbersArePoints
* EigenValue 
* EigenVector 
* LinearAlgebra 
* SolvingLinearEquations 
* MatrixRank
* MatrixDeterminant 
* MatrixInverse
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CategoryMath