Intuition behind speciality of symmetric matrices What is the geometric intuition behind the fact that only matrices that are similar to a symmetric matrix are diagonizable? 
So e.g. why is it important that the multiplier of the the first component of the last basis vector be the same as the multiplier of the the last component of the first basis vector(i.e. that in an $n*n $ matrix $(n,1)$ be the same as $(1,n)$)?


only matrices that are similar to a symmetric matrix are diagonizable

This statement is true, but quite useless. I prefer the statement

only matrices that are similar to a diagonal matrix are diagonizable

which is, hum... the definition itself of diagonalizability. The geometrical intuition of diagonalizability is that you can decompose a transformation in homotheties, which are the simplest geometrical transformations you could imagine.
