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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?

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One good way to think about a matrix is by its eigenvalues an eigenvectors.

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 $\sqrt{2}$.

Another helpful tool here is singular value decomposition.

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  • $\begingroup$ I am only in high school so I have some trouble understanding your links. But thanks! $\endgroup$ – Arcthor Feb 14 '15 at 23:22
  • $\begingroup$ 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. $\endgroup$ – Omnomnomnom Feb 14 '15 at 23:25
  • $\begingroup$ Yeah sorry, my bad. I have an assignment that reads "describe all changes to a geographical area transformed by the matrix (given in OP)" and I don't really know how to proceed or what the teacher is after. Does it rotate? Is it reflected on the x-axis etc. $\endgroup$ – Arcthor Feb 14 '15 at 23:33
  • $\begingroup$ 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 $(0,1)$. $\endgroup$ – Omnomnomnom Feb 14 '15 at 23:41
  • $\begingroup$ Okay I understand a bit better when plugging in (1,0) and (0,1), thank you. But is there any reason to believe that the transformation will have another result, other than rotation by 135 degrees and scaling by sqrt(2), if applied to for example (234, 80)? $\endgroup$ – Arcthor Feb 14 '15 at 23:58

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