I know that the matrix representation of the linear transformation: x 1 </msub>

George Bray

George Bray

Answered question

2022-06-16

I know that the matrix representation of the linear transformation:
x 1 = X 1 + 2 λ X 2
x 2 = X 2 λ X 1
is:
[ x 1 x 2 ] = [ 1 2 λ λ 1 ] [ X 1 X 2 ]
but what if I have the non-linear transformation:
x 1 = X 1 + 2 λ X 2 X 1 2
x 2 = X 2 λ X 1
How can it be expressed in matrix form to get it's inverse?

Answer & Explanation

Amy Daniels

Amy Daniels

Beginner2022-06-17Added 20 answers

If λ = 0, the transformation is clearly a bijection (geometrical interpretation : it is a symmetry with respect to the line with equation y = x),For any nonzero value of λ, this transformation cannot be a bijection because point (0,2) would be the image of 3 different points by the inverse transformation. These points are:
(1) ( X 1 , X 2 ) = { ( 0 , 2 )   it's a fixed point. ( 1 λ ( 1 + a ) , 1 + a ) ( 1 λ ( 1 a ) , 1 a )       with       a := 2 2
Using other words, your system taken with x 1 = 0 and x 2 = 2, i.e.,:
(2) { ( i ) . . . 0 = X 1 + 2 λ X 2 X 1 2 ( i i ) . . . 2 = X 2 λ X 1
has the 3 solutions given in (1).
Jaqueline Kirby

Jaqueline Kirby

Beginner2022-06-18Added 6 answers

Remarks to exparet's answer:
- The second and third solutions in (1) are easily found by expressing in (2)(ii) X 2 = λ X 1 + 2 and then plugging this expression into (2)(i), getting a quadratic equation for X 1 .
- All points ( 0 , y ) are fixed points of the transformation.
- There are many cases where, being given ( x 1 , x 2 ), there are two solutions for ( X 1 , X 2 ).

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