Invariant points and lines

Objective. Find geometric structures unchanged by transformations.

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Learn: the key idea

An invariant point of a transformation matrix M satisfies Mx = x, so it is unchanged; an invariant line is a line whose points, though possibly moved along it, always map back onto the same line. Invariant points other than the origin occur when M has eigenvalue 1, found by solving (M − I)x = 0. Invariant lines through the origin occur for any eigenvalue, since Mx = λx keeps the direction fixed even if points move along the line. For example, a shear matrix [[1,2],[0,1]] fixes every point on the x-axis, making it a line of invariant points.

Worked example

Find the invariant lines through the origin for the matrix M = [[3,0],[0,1]], giving their equations.

1 $M is diagonal, so its eigenvalues are its diagonal entries: 3 and 1$
2 $Eigenvalue 3 has eigenvector direction (1,0), giving the line y = 0$
3 $Eigenvalue 1 has eigenvector direction (0,1), giving the line x = 0$
4 $Both lines map to themselves under M, so both are invariant$

Answer: y = 0 and x = 0

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Common mistake. Students confuse an invariant line with a line of invariant points, assuming every point on an invariant line is fixed. Only lines of eigenvalue 1 have every point fixed; other invariant lines only map onto themselves as a set.

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