4
$\begingroup$

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 ?

  • 0
    Can you clarify "Now, they ask me which type of application is ff, the application given by the matrix AA, but I don't know."? – 2017-02-09
  • 0
    @Test123: I mean, what represent geometrically $A$ ? Is it a rotation ? a translation ? A specific isometry ? Or anything else ? – 2017-02-09

1 Answers 1

3

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 $25$".