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Let us denote two sets of matrices: \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes the spectral radius of a matrix and \begin{align*} \mathcal H = \{A \in M_n(\mathbb R): \max_{i} \text{Re}(\lambda_i(A)) < 0\}, \end{align*} i.e., matrices with eigenvalues lying on the left open half plane. It is not hard to see the transformation $f: \mathcal S \to \mathcal H$ given by $A \mapsto (A-I)^{-1}(A+I)$ is well-defined and indeed a diffeomorphism.

I am trying to determine whether $f$ preserves the order of Krylov subspace. More precisely, let $A \in \mathcal S$, $v$ be a vector and $\text{Krov}(A, v) = \{v, Av, \dots, A^r v\}$, where $\{v, Av, \dots, A^rv\}$ is a linearly independent set. Now would $\{v, f(A)v, \dots, [f(A)]^r v\}$ be a linearly independent set in general?

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