TITLE: What is the geometrical interpretation of $f(x,y)=\begin{pmatrix}9&12\\12&16\end{pmatrix}\begin{pmatrix}x\\y\end{pmatrix}$
QUESTION [5 upvotes]: Consider the matrix $A=\begin{pmatrix}9&12\\12&16\end{pmatrix}$. I show that $\begin{pmatrix}-4\\ 3\end{pmatrix}$ is a eigenvector or the eigen value $0$ and that $\begin{pmatrix}3\\ 4\end{pmatrix}$ is an eigenvector for the eigenvalue $25$. Therefore, in the basis $\left\{\begin{pmatrix}-4\\ 3\end{pmatrix},\begin{pmatrix}3\\ 4\end{pmatrix}\right\}$, is given by $$\begin{pmatrix}0&0\\0&25\end{pmatrix}.$$
Now, they ask me which type of application is $f$, the application given by the matrix $A$, but I don't know. I would say something like a projection on $Span\{(3,4)\}$ and an homothetic transformation in the direction of center $(0,0)$ and of parameter $\lambda=25$, is it correct ? Can we do better ?

REPLY [3 votes]: I think your description is pretty much as good as we can do.  I'd say "$f$ is an orthogonal projection onto the span of $(3,4)$ followed by a scaling by a factor of $25$".