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Invariant points and lines of a matrix

How do you find the invariant points of a matrix?
I've got the matrix
( 3 2)
( 1 2)
(2x2 matrix rows 32 and 12)
And I think one of the invariant lines is y=-x
Is this right and are there any others?

(edited 4 years ago)

Reply 1

ThiagoBrigido

Original post by Student135246

How do you find the invariant points of a matrix?
I've got the matrix
( 3 2)
( 1 2)
(2x2 matrix rows 32 and 12)
And I think one of the invariant lines is y=-x
Is this right and are there any others?

Do you know the difference between different lines and invariant points?

Reply 2

Student135246

Original post by ThiagoBrigido

Do you know the difference between different lines and invariant points?

i think invariant points are points that map onto themselves
invariant lines are lines where any point on that line will be mapped onto the line
is this right?

Reply 3

RDKGames

Study Forum Helper

Original post by Student135246

How do you find the invariant points of a matrix?
I've got the matrix
( 3 2)
( 1 2)
(2x2 matrix rows 32 and 12)
And I think one of the invariant lines is y=-x
Is this right and are there any others?

Let's check, if I take any point on the line

y=-x

and apply the matrix to it, I should get back to the original point, i.e.

\begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ -x \end{pmatrix} = \begin{pmatrix} x \\ -x \end{pmatrix}

must be satisfied. Which it is, just check by expanding the LHS.

Generally, you find invariant lines by seeking them in the form

y=mx

and determining what

m

must be.

Imposing this, you simply need to solve;

\begin{pmatrix} 3 & 2 \\ 1 & 2 \end{pmatrix} \begin{pmatrix} x \\ mx \end{pmatrix} = \begin{pmatrix} x \\ mx \end{pmatrix}

Expanding the top row;

3x + 2mx = x \implies (2+2m)x = 0 \implies m = -1

and this is the ONLY line, because there are not other solutions to this equation. You can check that this satisfies the expansion of the bottom row;

x + 2mx = mx

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