Given a linear transformation, a coordinate transformation to the eigenbasis (if it exists) seems the most optimal and easiest to calculate. An eigenvector expressed in other coordinates always seems to be more complicated to write. For example for an eigenvector ${\bf v}_1$ in eigenbasis we have:
$$ {\bf v}_1 = 1 {\bf v}_1 $$
which seems trivial. But if we use a arbitrary different coordinate system we get
$$ {\bf v}_1 = c_1 {\bf w}_1 + c_2 {\bf w}_2 + \dots + c_n {\bf w}_n $$
It always seems you need more basis vectors to express this eigenvector than in its eigenbasis. Is this true?
Can you define a 'most' optimal coordinate system and are there any theorems about which are the most 'optimal'?