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**Extra resources for Linear system theory : the state space approach**

**Example text**

The classes of structured matrices we consider in this paper are the following. Throughout the paper let A denote either the transpose AT or the conjugate transpose A∗ of a matrix A. Furthermore, let H = H ∈ Fn×n and −J T = J ∈ Fn×n be invertible. Then A ∈ Fn×n is called 1. H-selfadjoint, if F = C, = ∗, and A∗ H = HA; 2. H-symmetric, if F ∈ {R, C}, = T , and AT H = HA; 3. J-Hamiltonian, if F ∈ {R, C}, = T , and AT J = −JA. There is no need to consider H-skew-adjoint matrices A ∈ Cn,n satisfying A∗ H = −HA, because this case can be reduced to the case of H-selfadjoint matrices by considering iA instead.

Nm ), and hence, the algebraic multiplicity aU of A + U RU G at λ must be greater than or equal to a0 := nk+1 + · · · + nm . However, by hypothesis there exists one U0 so that A + U0 RU0 G has exactly the partial multiplicities (nk+1 , . . , nm ), so in particular it has the algebraic multiplicity aU0 = a0 . Therefore, by (2) the set Ω1 of all U ∈ Cn×k satisfying aU = a0 is generic and for all U ∈ Ω1 . Since (nk+1 , . . , nm ) is the only possible list of partial multiplicities that dominates (nk+1 , .

Am , where aj ≤ aj for j = 1, . . , m. Furthermore, let ε > 0 be such that the discs Dj := μ ∈ C |λj − μ| < ε2/n , j = 1, . . , m are pairwise disjoint. If for each j = 1, . . , m there exists a matrix Uj ∈ Cn×k with Uj < ε such that the matrix A + Uj RUj G has exactly (aj − aj ) simple eigenvalues in Dj diﬀerent from λj , then generically (with respect to the entries of U if = T and with respect to the real and imaginary parts of the entries of U if = ∗) the eigenvalues of A + U RU G that are diﬀerent from the eigenvalues of A are simple.