Solutions to a quadratic inverse eigenvalue problem

Abstract In this paper, we consider the quadratic inverse eigenvalue problem (QIEP) of constructing real symmetric matrices M , C , and K of size n × n , with ( M , C , K ) ≠ 0 , so that the quadratic matrix polynomial Q ( λ ) = λ 2 M + λ C + K has m ( n m ⩽ 2 n ) prescribed eigenpairs. It is shown that, for almost all prescribed eigenpairs, the QIEP has a solution with M nonsingular if m m ∗ , and has only solutions with ( Q ( λ ) ) ≡ 0 otherwise, where m ∗ = n + ( 1 + 1 + 8 n ) / 2 . We also derive the expression of the general solution of the QIEP for both cases. Furthermore, we develop an algorithm for finding a particular solution to the QIEP with M positive definite if it exists.

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