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?

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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&#039;s inverse?

Amy Daniels · Accepted Answer

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&#039;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 · Accepted Answer

Remarks to exparet&#039;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    ).

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

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