I'm trying to solve problem 3.2.5 from the book A First Course in Dynamics of B. Hasselblatt and A. Katok that says:
Suppose a linear map with a double negative eigenvalue arises from the solution of a linear differential equation. Show that it is proportional to the identity.
I am assuming that I should take a $2\times 2$ matrix because the problem is from chapter 3: Linear Maps and Linear Differential Equations, that develops all content using $2\times 2$ matrices and because no other information about other posible eigenvalues is given.
Taking the $2\times 2$ matrix $$A = \begin{bmatrix} a & b\\\ c & d \end{bmatrix},$$ if $A$ has a repetead negative eivenlue $\lambda$, I can conclude from the characteristic polynomial that $\big[\mathrm{Tr}(A)\big]^2 - 4\det(A)$ must be cero, so $\lambda = \mathrm{Tr}(A)/2$ with $\mathrm{Tr}(A) < 0$.
But this is not enought to ensure that $A$ will be a diagonal matrix, and therefore, a matrix proportional to de identity. I can find non-diagonal $2\times 2$ matrices that have a reapeated negative eigenvalue. (In fact, I need to conclude something stronger: $a = d < 0$ and $b = c = 0$).
But I don't know how to obtain more assumptions to lead to those conclusions. I guess that it has someting to do with the part of the problem that says "Suppose a linear map [...] arises from the solution of a linear differential equation." But I don't get what that is saying to me. It implies that the matrix must be diagonlizable or something like that?
Thank you in advance.