# Determining what a transformation matrix does

How do I know what a transformation matrix actually does, without just testing it?

Is there any general way to determine what a transformation matrix, like

$$\begin{matrix}-1 & -1 \\ 1 & -1 \end{matrix}$$ does?

In this case, we see that the matrix has eigenvalues $-1 \pm i = \sqrt{2} e^{ \pm 3\pi i/4 }$. It follows that this matrix is a rotation by angle $3 \pi/4$, followed by dilation by factor $\sqrt{2}$.
• That would have been helpful information to add to the question. At any rate, for a better answer, you should give us a better idea of what you mean by "what a transformation matrix actually does". For example, I could say that this matrix "actually" takes the vector $(x,y)$ to the vector $(-x-y,x-y)$; but somehow that doesn't seem very helpful. – Omnomnomnom Feb 14 '15 at 23:25
• Well, as I said, this one is a rotation by $135 \circ$ in some direction followed by a scaling by $\sqrt 2$. You should plug in $(1,0)$ and $(0,1)$ to figure out which direction things are being rotated. In general, you get a pretty good idea of what's going on by plugging in $(1,0)$ and $(0,1)$. – Omnomnomnom Feb 14 '15 at 23:41