There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:

1. All eigenvalues of $A$ have negative real parts
2. For any symmetric, positive definite matrix $Q$ there is a unique symmetric, positive definite solution $P$ to the Lyapunov equation $A^T P + PA = -Q$
3. The equilibrium of $\dot{x} = Ax$ is exponentially stable

All these statements in some way or another use the matrix exponential $e^{At}$. In particular, the solution to the Lyapunov equation can be found by $P=\int \limits_{0}^{\infty} e^{A^Tt}Qe^{At} dt$. In turn, solutions to the ODE have the from $e^{At}x_0$.

All proofs I know that show equivalence of these three statements use the Jordan normal form of $A$. In fact, the matrix exponential itself is notoriously hard to handle in practice, and one tries to represent it in some suitable form via the said normal form, or by partial fraction decomposition of $sI - A$ and inverse Laplace transforming it.

My question is: do there exist proofs of equivalence of the said three statements (or some of them) that do not use the Jordan normal form and possibly do not explicitly use eigenvectors?

The motivation of my question is the following: computation of eigenvectors is computationally unstable, whereas eigenvalues are stable. In practice, one can compute eigenvalues and check their real parts. However, it seems computationally intractable to find exact analytic forms of solutions of linear homogeneous ODE's in general. I was thinking of some computational apparatus for stability that wouldn't resort to eigenvectors.

Update

I have an idea how to go about this problem. In order for that to work, I require a bound on the condition number of $V$ of the Jordan normal form $B = V^{-1} J V$ for some $B$ depending on some bound on the elements of $B$, or, say, its spectral norm. Does there exists such a bound? The point is, in bounding the matrix exponential, the condition number of $V$ arises.

There are some equivalent statements in the classical stability theory of linear homogeneous differential equations $ \dot{x} = Ax, x \in \mathbb{R}^n $ such as:

1. All eigenvalues of $A$ have negative real parts
2. For any symmetric, positive definite matrix $Q$ there is a unique symmetric, positive definite solution $P$ to the Lyapunov equation $A^T P + PA = -Q$
3. The equilibrium of $\dot{x} = Ax$ is exponentially stable

All these statements in some way or another use the matrix exponential $e^{At}$. In particular, the solution to the Lyapunov equation can be found by $P=\int \limits_{0}^{\infty} e^{A^Tt}Qe^{At} dt$. In turn, solutions to the ODE have the from $e^{At}x_0$.

All proofs I know that show equivalence of these three statements use the Jordan normal form of $A$. In fact, the matrix exponential itself is notoriously hard to handle in practice, and one tries to represent it in some suitable form via the said normal form, or by partial fraction decomposition of $sI - A$ and inverse Laplace transforming it.

My question is: do there exist proofs of equivalence of the said three statements (or some of them) that do not use the Jordan normal form and possibly do not explicitly use eigenvectors?