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In mathematics, the quadratic eigenvalue problem , is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that
where Q = λ 2 A 2 + λ A 1 + A 0 {\displaystyle Q=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}} , with matrix coefficients A 2 , A 1 , A 0 ∈ C n × n {\displaystyle A_{2},\,A_{1},A_{0}\in \mathbb {C} ^{n\times n}} and we require that A 2 ≠ 0 {\displaystyle A_{2}\,\neq 0} ,. There are 2 n {\displaystyle 2n} eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem. Q {\displaystyle Q} is also known as a quadratic polynomial matrix.