An inverse problem for gyroscopic systems

A “gyroscopic system” is a Hermitian matrix-valued function of the form L(λ)=Mλ2+iGλ+CL(λ)=Mλ2+iGλ+C where M,G,C∈Rn×nM,G,C∈Rn×n with M>0M>0 (positive definite), GT=−G≠0GT=−G≠0, CT=CCT=C and may be indefinite. Here we study factorizations of the form L(λ)=(Iλ−B)M(Iλ−A)L(λ)=(Iλ−B)M(Iλ−A), where A,B∈Cn×nA,B∈Cn×n, and use them to construct gyroscopic systems with specified right divisor Iλ−AIλ−A. We examine the constraints on the choice of A.