How would one interpret eigenvalues and eigenvectors? I tried googeling, but could find anything concrete. If i have an matrix consisting of the basis vectors x((3,1),(2,2)) for $\mathbb{R}^4$ and to eigenvalues 4 and -4. What does this mean geometric? I am looking for an intuitive way to visualise it.

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    [How to intuitively understand eigenvalue and eigenvector?](http://math.stackexchange.com/questions/243533/how-to-intuitively-understand-eigenvalue-and-eigenvector), [What is the importance of eigenvalues/eigenvectors?](http://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors), [What exactly are eigen-things?](http://math.stackexchange.com/questions/300145/what-exactly-are-eigen-things), [A simple explanation of eigenvectors...](http://math.stackexchange.com/questions/36815/a-simple-explanation-of-eigenvectors-and-eigenvalues-with-big-picture-ideas-of) –  Nov 26 '16 at 23:13
  • But the short answer is that eigenvectors are vectors whose direction stays the same under the action of the matrix. For instance $(0,1)^T$ maps to $(0,2)^T$ under $\pmatrix{1 & 0 \\ 1 & 2}$. Because $(0,1)^T$ is in the same direction as $(0,2)^T$, it means that $(0,1)^T$ is an eigenvector of the matrix. –  Nov 26 '16 at 23:16

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