I am trying to get an intuitive understanding of the meanings of right and left eigenvectors. I guess the best thing you can do is to provide examples of application. (Examples from the field of biology would be especially welcome).
Example:
A population have individuals that are classified in three classes, class A, B and C. Na, Nb, Nc represents the number of individuals in each class.
$$Na(t+1) = m*Na(t) + n*Nb(t) + o*Nc(t)$$
$$Nb(t+1) = p*Na(t) + q*Nb(t) + r*Nc(t)$$
$$Nc(t+1) = s*Na(t) + t*Nb(t) + u*Nc(t)$$
$$v(t+1) = A * v(t)$$
What is the long term equilibrium of this system of equations? Do we use left or right leading eigenvector of A. Why would be the biological meaning (if any) of the other (left/right) eigenvector?
The current intuitive sense I have is that an eigenvector of a matrix is a measure of how oriented is the distortion caused by the multiplication by this matrix. The eigenvalue is the strength of this distortion. Therefore the eigenvector linked with the biggest eigenvalue determines the long term behaviour of a system.
Here are two links on Stack exchange that did not help me answering my question.
How to intuitively understand eigenvalue and eigenvector?
question on left and right eigenvectors
Best Answer
I suppose in the question $v(t)$ just is an abbreviation for the column vector with components $Na(t), Nb(t), Nc(t)$, which describes the distribution among classes in the population. As you can see the equation lets the matrix $A$ operate from the left on this column vector (as it should; on the other side a row vector would be required), so the interesting case would be when $v(t)$ is a right eigenvector for $A$. In that case the operation by$~A$ multiplies all components by the same scalar, which means one gets the same relative distribution in classes as before; all classes have grown by the same factor. This means it is a relative equilibrium situation (though maybe an unstable one). It would seem to me that the situation can only be sustainable if the eigenvalue equals$~1$, since otherwise one would have exponential growth, which is very hard to sustain (unless it is by a factor less than$~1$, in which case it leads to extinction).
In this setting I can see no obvious use for left eigenvectors, since they do not correspond to particular states of the system (distributions into classes) like equilibrium states, but rather to linear functions on the set of all possible states. Such a function might measure something like the amount of certain resources that would be required by the population, as depending on the state the population.