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?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.