## On inverse eigenvalue problems of quadratic palindromic systems with partially prescribed eigenstructure.(English)Zbl 1427.65051

Summary: The palindromic inverse eigenvalue problem (PIEP) of constructing matrices $$A$$ and $$Q$$ of size $$n \times n$$ for the quadratic palindromic polynomial $$P(\lambda) = \lambda^2 A^{\star} + \lambda Q + A$$ so that $$P(\lambda)$$ has $$p$$ prescribed eigenpairs is considered. This paper provides two different methods to solve PIEP, and it is shown via construction that PIEP is always solvable for any $$p (1 \leq p \leq (3n+1)/2)$$ prescribed eigenpairs. The eigenstructure of the resulting $$P(\lambda)$$ is completely analyzed.

### MSC:

 65F18 Numerical solutions to inverse eigenvalue problems 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 15A24 Matrix equations and identities 15A29 Inverse problems in linear algebra

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### References:

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