Linear Transformation

An $n\times n$ Matrix ${\hbox{\sf A}}$ is a linear transformation (linear Map) Iff, for every pair of $n$-Vectors ${\bf X}$ and ${\bf Y}$ and every Scalar $t$,

$${\hbox{\sf A}}({\bf X}+{\bf Y}) = {\hbox{\sf A}}({\bf X}) + {\hbox{\sf A}}({\bf Y})$$ (1)

and

$${\hbox{\sf A}}(t{\bf X}) = t{\hbox{\sf A}}({\bf X}).$$ (2)

Consider the 2-D transformation

$$\rho x_1' = a_{11}x_1+a_{12}x_2$$ (3)

$$\rho x_2' = a_{21}x_1+a_{22}x_2.$$ (4)

Rescale by defining $\lambda\equiv x_1/x_2$ and $\lambda'\equiv x_1'/x_2'$, then the above equations become

$$\lambda'={\alpha \lambda+\beta\over\gamma\lambda+\delta},$$ (5)

where $\alpha\delta-\beta\gamma\not=0$ and $\alpha$, $\beta$, $\gamma$ and $\delta$ are defined in terms of the old constants. Solving for $\lambda$ gives

$$\lambda={\delta\lambda'-\beta\over -\gamma\lambda'+\alpha},$$ (6)

so the transformation is One-to-One. To find the Fixed Points of the transformation, set $\lambda=\lambda'$ to obtain

$$\gamma\lambda^2+(\delta-\alpha)\lambda-\beta=0.$$ (7)

This gives two fixed points which may be distinct or coincident. The fixed points are classified as follows.

variables	type
$(\delta-\alpha)^2+4\beta\gamma>0$	Hyperbolic Fixed Point
$(\delta-\alpha)^2+4\beta\gamma<0$	Elliptic Fixed Point
$(\delta-\alpha)^2+4\beta\gamma=0$	Parabolic Fixed Point

See also Elliptic Fixed Point (Map), Hyperbolic Fixed Point (Map), Involuntary, Linear Operator, Parabolic Fixed Point

References

Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 13-15, 1961.