EKElizaveta A. Kalinina
Elizaveta A. Kalinina
*This is an algorithmically generated author record based on publicly available information.
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Elizaveta A. Kalinina has a total of 2 co-authors. Here are the top co-authors by number of shared publications:
- Alexei Yu. Uteshev
St. Petersburg State University
1 shared publication Marina V. Goncharova
Faculty of Applied Mathematics St St. Petersburg State University Petersburg Russia
1 shared publication
Latest Publications
- Open Access
Distance Evaluation to the Set of Defective Matrices
We treat the problem of the Frobenius distance evaluation from a given matrix $ A \in \mathbb R^{n\times n} $ with distinct eigenvalues to the manifold of matrices with multiple eigenvalues. On restricting considerations to the rank $ 1 $ real perturbation matrices, we prove that the distance in question equals $ \sqrt{z_{\ast}} $ where $ z_{\ast} $ is a positive (generically, the least positive) zero of the algebraic equation $$ \mathcal F(z) = 0, \ \mbox{where} \ \mathcal F(z):= \mathcal D_λ \left( \det \left[ (λI - A)(λI - A^{\top})-z I_n \right] \right)/z^n $$ and $ \mathcal D_λ $ stands for the discriminant of the polynomial treated with respect to $λ$. In the framework of this approach we also provide the procedure for finding the nearest to $ A $ matrix with multiple eigenvalue. Generalization of the problem to the case of complex perturbations is also discussed. Several examples are presented clarifying the computational aspects of the approach.Open AccessDistance Evaluation to the Set of Defective Matrices
We treat the problem of the Frobenius distance evaluation from a given matrix $ A \in \mathbb R^{n\times n} $ with distinct eigenvalues to the manifold of matrices with multiple eigenvalues. On restricting considerations to the rank $ 1 $ real perturbation matrices, we prove that the distance in question equals $ \sqrt{z_{\ast}} $ where $ z_{\ast} $ is a positive (generically, the least positive) zero of the algebraic equation $$ \mathcal F(z) = 0, \ \mbox{where} \ \mathcal F(z):= \mathcal D_λ \left( \det \left[ (λI - A)(λI - A^{\top})-z I_n \right] \right)/z^n $$ and $ \mathcal D_λ $ stands for the discriminant of the polynomial treated with respect to $λ$. In the framework of this approach we also provide the procedure for finding the nearest to $ A $ matrix with multiple eigenvalue. Generalization of the problem to the case of complex perturbations is also discussed. Several examples are presented clarifying the computational aspects of the approach.