Solutions to a quadratic inverse eigenvalue problem

General Solutions for a Class of Inverse Quadratic Eigenvalue Problems

“…To solve the ISQEP (n + 1 ≤ k ≤ 2n), we cite the following lemma [2], and then obtain the general solution of the ISQEP in a parameterized form.…”

“…Substituting (2.14) into (2.13), and using assumption (2) and the same technique as in Ref. [19], we obtain that Γ i j = 0 for j = i, Γ j j λ [2] j − (λ [2] j ) T Γ j j = 0 , j = 1, 2, · · · , l (2.15) and Γ l+ j,l+ j λ 2l+ j − λ 2l+ j Γ l+ j,l+ j = 0 , j = 1, 2, · · · , s − l .…”

“…Since λ [2] j has the form in (1.4) with β j = 0, it is easy to see that the general solution of (2.5) has the form…”

“…Consider the ISQEP where the partial eigen-structure (Λ, X ) ∈ 5×5 × 5×5 is as in (1.3) and (1.5), with λ 1 = −0.31828 − 0.86754i =λ 2 , λ 3 = −0.95669 + 0.17379i = λ 4 , λ 5 = −4.4955 , and the corresponding eigenvectors It is easy to check that the matrix pair (Λ, X ) ∈ 5×5 × 5×5 satisfies the assumptions (1) and (2).…”

Semi-definite programming techniques for structured quadratic inverse eigenvalue problems

“…With only partially prescribed eigenpairs available, Chu, Kuo and Lin [16] put forward a special solution for the symmetric QIEP, guaranteeing M ≻ 0, K ⪰ 0. Shortly afterwards, a sufficient condition for the general solution parameterized in terms of the QR decomposition of the eigenvectors was developed by Kuo, Lin and Xu [29] for the case k ≤ n and by Cai, Kuo, Lin and Xu [11] for the case k > n . The feedback control approaches proposed in [19, 20, 35], on the other hand, generally can maintain symmetry only.…”