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FP1 Invariant Points

Hi, I was wondering if anyone could help me with finding invariant points. The question is to find the invariant points for transformations of the following matrix:

(0−112)\begin{pmatrix} 0 & -1 \\ 1 & 2 \end{pmatrix}

Here's what I did:

(0−112)\begin{pmatrix} 0 & -1 \\ 1 & 2 \end{pmatrix} (xy)\begin{pmatrix} x \\ y \end{pmatrix} = (xy)\begin{pmatrix} x \\ y \end{pmatrix}

(−yx+2y)\begin{pmatrix} -y \\ x+2y \end{pmatrix} = (xy)\begin{pmatrix} x \\ y \end{pmatrix}

−x=x+2y-x = x+2y
−2x=2y-2x=2y
−x=y-x=y

This is the only method I can figure out of solving it, and as far as I can tell I've made no mistakes, but then the answer given in the book is (λ\lambda, -λ\lambda), and I can't for the life of me figure out what Lambda is supposed to represent! Can somebody please explain this, and if I've not solved this properly, I'd really appreciate a step-by-step solution if anyone can spare the time. Thanks in advance! <3 xoxo
Reply 1
Avatar for Hasufel
Hasufel
16
take the corresponding entries in the top "row" - −y=x-y=x OR: y=−xy=-x which implies that the solution is points of the form

(x,−x)(x,-x) or, as they put it: (λ,−λ)(\lambda, -\lambda)

(the lower rows equated just confirms the top one, once you sub y=-x into it)

they just use lambda as it is arbitrary, to let you see that the solution is any one number, the y of which is the negation.
Original post by mattallica
Hi, I was wondering if anyone could help me with finding invariant points. The question is to find the invariant points for transformations of the following matrix:

(0−112)\begin{pmatrix} 0 & -1 \\ 1 & 2 \end{pmatrix}

Here's what I did:

(0−112)\begin{pmatrix} 0 & -1 \\ 1 & 2 \end{pmatrix} (xy)\begin{pmatrix} x \\ y \end{pmatrix} = (xy)\begin{pmatrix} x \\ y \end{pmatrix}

(−yx+2y)\begin{pmatrix} -y \\ x+2y \end{pmatrix} = (xy)\begin{pmatrix} x \\ y \end{pmatrix}

−x=x+2y-x = x+2y
−2x=2y-2x=2y
−x=y-x=y

This is the only method I can figure out of solving it, and as far as I can tell I've made no mistakes, but then the answer given in the book is (λ\lambda, -λ\lambda), and I can't for the life of me figure out what Lambda is supposed to represent! Can somebody please explain this, and if I've not solved this properly, I'd really appreciate a step-by-step solution if anyone can spare the time. Thanks in advance! <3 xoxo


λ \lambda just indicates that it is a line of invariant points. You could just have correctly given it as (k,-k) or (t,-t) the letter used is arbitrary.
Reply 3
Avatar for mattallica
mattallica
OP
1
Original post by brianeverit
λ \lambda just indicates that it is a line of invariant points. You could just have correctly given it as (k,-k) or (t,-t) the letter used is arbitrary.


So if I was to say (λ\lambda, λ\lambda) would that just be the same as x=y?
Reply 4
Avatar for davros
davros
16
Original post by mattallica
So if I was to say (λ\lambda, λ\lambda) would that just be the same as x=y?


Yes :smile:

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