On existence of the solutions of inhomogeneous eigenvalue problem (Q1179946)

For a given real \(n\times n\)-matrix \(A\), given \(s>0\), \(b\in\mathbb{R}^ n\) a pair \((x,\lambda)\), \(x\in\mathbb{R}^ n\), \(\lambda\in R\) is called an inhomogeneous eigenvector and inhomogeneous eigenvalue, if \(Ax=\lambda x+b\), \(x^ Tx=s^ 2\) holds. The following result, complementing results of \textit{R. M. M. Mattheij} and \textit{G. Söderlind} [Linear Algebra Appl. 88/89, 507-531 (1987; Zbl 0623.65039)], is proved: If for some \(i \) \(b^ 2_ i>s^ 2\sum_ j a^ 2_{ij}\) then there exist at least two inhomogeneous eigenvalues. The proof uses a fixed point argument.