Matrix $M$ has a determinant value = $d$.
If Matrix $M$ that has an eigenvector $V$ and corresponding eigenvalue $\lambda$,
Then is there another Matrix $M'$ with determinant value of $d$, that can transform vector $V$ by an amount greater than or equal to $\lambda$ ?
In simpler terms, can a determinant be viewed as a limited resource that represents the cumulative power of translation of rotation of the matrix, when used to transform a particular vector ?
In that case, can we view a matrix to be using all of its limited transformative power for scaling the eigenvector, thus achieving the maximum scaling of the same eigenvector that can be achieved by any other matrix with the same determinant ?
Thank you.
edit:
I will try to state the problem again, hope it helps.
$V$ is an Eigenvector of $M$. $V$ is not an Eigenvector of $M′$. Both $M$ and $M′$ have the same determinant equal to $d$. Is there an $M′$ such that it scales (by transforming) $V$ by a value greater than λ? (The extent of rotation is irrelevant to the problem)
edit 2 :
Read the reply more carefully. Given your answer, I think my suggested intuition for Eigenvalues doesn't hold. My suggested intuition seems to be straight up incorrect.