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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})
\end{displaymath} (1)

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

Consider the 2-D transformation

$\displaystyle \rho x_1'$ $\textstyle =$ $\displaystyle a_{11}x_1+a_{12}x_2$ (3)
$\displaystyle \rho x_2'$ $\textstyle =$ $\displaystyle a_{21}x_1+a_{22}x_2.$ (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},
\end{displaymath} (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},
\end{displaymath} (6)

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


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

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© 1996-9 Eric W. Weisstein