FPRs of rational matrices

As a consequence of Theorem 6.2.4 and Theorem 5.5.8, we have the following result for minimal bases and minimal indices of G(λ).

Theorem 6.2.5. Consider the PGF pencil Tω(λ) := λM^S−ω₁ −M^Sω0 of G(λ) associ- ated with a proper permutation ω := (ω₀, ω₁) of {0,1, . . . , m}. Suppose that CIP(ω₀) = (c₀, i₀). Let ω₁ be given by ω₁ := (σ₁, m, σ₂). Let i_T := i(rev(σ₁), ω₀, rev(σ₂)) and c_T := c(rev(σ₁), ω₀, rev(σ₂)) be the total number of inversions and consecutions of (rev(σ₁), ω₀, rev(σ₂)), respectively. Let w_i(λ) :=



 u_i(λ) v_i(λ)



 ∈ C[λ]^mn+r, where u_i(λ) ∈ C[λ]^mn and vi(λ)∈C[λ]^r for i= 1 : p. Then we have the following.

(a) Right minimal bases. If w₁(λ), . . . , w_p(λ)

is a right minimal basis of T^ω(λ) then

(

(e^T_m−c

0⊗I_n)u₁(λ), . . . ,(e^T_m−c

0⊗I_n)u_p(λ)

)

is a right minimal basis ofG(λ).Further, if ε₁ ≤ · · · ≤ε_p are the right minimal indices of Tω(λ) then ε₁−i_T ≤ · · · ≤ ε_p−i_T are the right minimal indices of G(λ).

(b) Left minimal bases. If w₁(λ), . . . , w_p(λ)

is a left minimal basis of Tω(λ) then

(

(e^T_m−i0⊗I_n)u₁(λ), . . . ,(e^T_m−i0⊗I_n)u_p(λ)

)

is a left minimal basis of G(λ). Further, if η₁ ≤ · · · ≤ η_p are the left minimal indices of Tω(λ) then η₁−c_T ≤ · · · ≤ η_p−c_T are the left minimal indices of G(λ).

Theorem 6.3.2. LetL(λ) := M^Sτ1M^Sσ1(λM^Sτ−M^Sσ)M^Sσ2M^Sτ2 andL(λ) := M_τ^P₁M_σ^P₁(λM_τ^P− M_σ^P)M_σ^P₂M_τ^P₂ be FPRs of G(λ) and P(λ), respectively. Then

L(λ) =





L(λ) em−i0(σ1,σ)⊗C e^T_m−c

0(σ,σ2)⊗B A−λE



.

Thus, the map FPR(P) −→ FPR(G), L(λ) 7−→





L(λ) em−i0(σ1,σ)⊗C e^T_m−c

0(σ,σ2)⊗B A−λE



 is a bijection, where FPR(P) and FPR(G) denote the set of FPRs of P(λ) and G(λ), respectively.

Proof. We will prove a more general result in Theorem 7.2.4.

We now describe the recovery of minimal bases and minimal indices of S(λ) from those of the FPRs ofS(λ). Recall from Definition 1.2.20 thatct(α) (resp.,it(α)) denotes the consecutions (resp., inversions) of an index tuple α at an index t.

Theorem 6.3.3. Let L(λ) := M^Sτ1M^Sσ1(λM^Sτ −M^Sσ)M^Sσ2M^Sτ2 be an FPR of S(λ). Let τ be given by τ := (τ_l,−m, τ_r). Set α := −rev(τ_l), σ,−rev(τ_r)

. Let c_L := c(α) and i_L := i(α) be the total number of consecutions and inversions of the permutation α of {0,1, . . . , m−1}. Then we have the following.

(I) Right minimal bases and right minimal indices.

(a) F^FPR_L (S) : N_r(L) → N_r(S),



 u(λ) v(λ)



 7→



 (e^T_m−c

0(σ,σ2)⊗I_n)u(λ) v(λ)



, is a linear isomorphism and maps a minimal basis of Nr(L) to a minimal basis of Nr(S), where u(λ) ∈ C(λ)^mn and v(λ) ∈C(λ)^r. Thus, if w₁(λ), . . . , w_p(λ)

is a right minimal basis of L(λ) then

(

_F^FPR_L (S)w₁(λ), . . . ,F^FPR_L (S)w_p(λ)

)

is a right minimal basis of S(λ).

(b) Ifε₁ ≤ · · · ≤ε_p are the right minimal indices of L(λ)then ε₁−i_L≤ · · · ≤ε_p−i_L are the right minimal indices of S(λ).

(II) Left minimal bases and left minimal indices.

(c) K^FPR_L (S) : N_l(L) → N_l(S),



 u(λ) v(λ)



 7→



 (e^T_m−i

0(σ1,σ)⊗I_n)u(λ) v(λ)



, is a linear iso- morphism and maps a minimal basis of Nl(L) to a minimal basis of Nl(S), where u(λ) ∈ C(λ)^mn and v(λ) ∈ C(λ)^r. Thus, if w₁(λ), . . . , w_p(λ)

is a left minimal ba- sis of L(λ) then

(

_K^FPR_L (S)w₁(λ), . . . ,K^FPR_L (S)w_p(λ)

)

is a left minimal basis of S(λ).

(d) If η₁ ≤ · · · ≤η_p are the left minimal indices ofL(λ)then η₁−c_L ≤ · · · ≤η_p−c_L are the left minimal indices of S(λ).

Proof. We haveL(λ) =M^Sτ1M^Sσ1T^ω(λ)M^Sσ2M^Sτ2, whereT^ω(λ) := λM^Sτ−M^Sσ is a PGF pen- cil ofG(λ) associated with the permutationω := (σ,−τ) of{0,1, . . . , m}. SinceM^Sσ2M^Sτ2

is nonsingular, it is easily seen that the map M^Sσ2M^Sτ2 : N_r(L) → N_r(T^ω), x(λ) 7→

M^Sσ2M^Sτ2

x(λ) is an isomorphism and maps a minimal basis ofN_r(L) to a minimal basis ofN_r(Tω). On the other hand, by Theorem 6.2.4,F^PGFω (S) :N_r(Tω)→ N_r(S),



 u(λ) v(λ)



7→



 (e^T_m−c

0(σ)⊗I_n)u(λ) v(λ)



is an isomorphism and maps a minimal basis ofN_r(T^ω) to a min- imal basis of N_r(S), where u(λ) ∈ C(λ)^mn and v(λ) ∈ C(λ)^r. Hence F^PGFω (S)M^Sσ2M^Sτ2 : N_r(L)→ N_r(S), y(λ)7→

(

_F^PGF_ω (S)M^Sσ2M^Sτ2

)

y(λ) is an isomorphism and maps a minimal basis of N_r(L) to a minimal basis of N_r(S). Now, by Lemma 2.1.11, we have

F^PGFω (S)M^Sσ2M^Sτ2 =



 (e^T_m−c

0(σ)⊗In)M_σ^P₂M_τ^P₂ I_r



=



 e^T_m−c

0(σ,σ2)⊗In

I_r



=F^FPR_L (S), which yields the desired result.

Now, let ε₁ ≤ · · · ≤ ε_p be the right minimal indices of L(λ). Since the PGF pencil Tω(λ) is strictly equivalent to L(λ), ε₁ ≤ · · · ≤ε_p are also the right minimal indices of Tω(λ). Hence by Theorem 6.2.4,ε₁−i_L≤ · · · ≤ε_p−i_L are the right minimal indices of S(λ).

For the recovery of left minimal bases, observe that (M^Sτ1M^Sσ1)^T :N_l(L)→ N_l(Tω), x(λ) 7→(M^Sτ1M^Sσ1)^Tx(λ) is an isomorphism and maps a minimal basis of N_l(L) to a minimal basis of N_l(Tω). Again by Theorem 6.2.4, K^PGFω (S) : N_l(Tω) → N_l(S),



 u(λ) v(λ)



 7−→



 (e^T_m−i

0(σ)⊗I_n)u(λ) v(λ)



is an isomorphism and maps a minimal basis ofN_l(Tω) to a min- imal basis ofN_l(S), whereu(λ)∈C(λ)^mnand v(λ)∈C(λ)^r. HenceK^PGFω (S)(M^Sτ1M^Sσ1)^T : N_l(L)→ N_l(S), y(λ)7→

(

_K^PGF_ω (S)(M^Sτ1M^Sσ1)^T

)

y(λ) is an isomorphism and maps a mini- mal basis of N_l(L) to a minimal basis of N_l(S). Now,

K^PGFω (S)(M^Sτ1M^Sσ1)^T =



 (e^T_m−i

0(σ)⊗I_n)(M_τ^P₁M_σ^P₁)^T I_r



=





M_τ^P₁M_σ^P₁(e_m−i0(σ)⊗I_n)T

I_r





and by Lemma 2.1.11, we have M_τ^P₁M_σ^P₁(em−i₀(σ)⊗I_n) = em−i₀(σ1,σ) ⊗I_n, which yields the desired result.

Finally, let η₁ ≤ · · · ≤ η_p be the left minimal indices of L(λ). Since the PGF pencil Tω(λ) is strictly equivalent to L(λ), η₁ ≤ · · · ≤ η_p are also the left minimal indices of

T^ω(λ). Hence by Theorem 6.2.4, η1−c_L ≤ · · · ≤ηp−c_L are the left minimal indices of S(λ). This completes the proof.

We have the following recovery rules for eigenvectors ofS(λ) when S(λ) is regular.

Theorem 6.3.4. Suppose that S(λ) is regular and µ∈C is an eigenvalue of S(λ). Let L(λ) := M^Sτ1M^Sσ1(λM^Sτ −M^Sσ)M^Sσ2M^Sτ2 be an FPR of S(λ). Then:

(a) Right eigenvectors. If



 u v



∈ Nr(L(µ)) then



 (e^T_m−c

0(σ,σ2)⊗I_n)u v



∈ Nr(S(µ)), where u ∈ C^mn and v ∈ C^r. In fact, the mapping



 u v



 7→



 (e^T_m−c

0(σ,σ2)⊗I_n)u v



 is a linear isomorphism between N_r(L(µ)) and N_r(S(µ)). Thus, if(w₁, . . . , w_p) is a basis of N_r(L(µ)) then

(

_F^FPR_L (S)w₁, . . . ,F^FPR_L (S)w_p

)

is a basis of N_r(S(µ)), where F^FPR_L is as given in Theorem 6.3.3.

(b) Left eigenvectors. If



 u v



 ∈ N_l(L(µ)) then



 (e^T_m−i

0(σ1,σ)⊗I_n)u v



 ∈ N_l(S(µ)), where u ∈ C^mn and v ∈ C^r. In fact, the mapping



 u v



 7→



 (e^T_m−i

0(σ1,σ)⊗I_n)u v



 is a linear isomorphism between N_l(L(µ)) and N_l(S(µ)). Thus, if (w₁, . . . , w_p) is a basis of N_l(L(µ))then

(

_K^FPR_L (S)w₁, . . . ,K^FPR_L (S)w_p

)

is a basis ofN_l(S(µ)), where K^FPR_L is as given in Theorem 6.3.3.

Proof. A verbatim proof of Theorem 6.3.3 yields the desired results.

As a consequence of Theorem 6.3.3 and Theorem 5.5.8, we have the following recovery rules for minimal bases and minimal indices of G(λ) from those of the FPRs ofG(λ).

Theorem 6.3.5. Let L(λ) := M^Sτ1M^Sσ1(λM^Sτ −M^Sσ)M^Sσ2M^Sτ2 be an FPR of G(λ). Let τ be given by τ := (τ_l,−m, τ_r). Set α := (−rev(τ_l), σ,−rev(τ_r)). Let c_L := c(α) and i_L := i(α) be the total number of consecutions and inversions of the permutation α of {0,1, . . . , m− 1}. Let wi(λ) :=



 u_i(λ) v_i(λ)



 ∈ C[λ]^mn+r, where ui(λ) ∈ C[λ]^mn and v_i(λ)∈C[λ]^r for i= 1 :p. Then we have the following.

(a) Right minimal bases. If w₁(λ), . . . , w_p(λ)

is a right minimal basis of L(λ) then

(

(e^T_m−c

0(σ,σ2)⊗I_n)u₁(λ), . . . ,(e^T_m−c

0(σ,σ2)⊗I_n)u_p(λ)

)

is a right minimal basis ofG(λ).

Further, ifε1 ≤ · · · ≤εp are the right minimal indices ofL(λ)thenε1−i_L ≤ · · · ≤εp−i_L are the right minimal indices of G(λ).

(b) Left minimal bases. If w1(λ), . . . , wp(λ)

is a left minimal basis of L(λ) then

(

(e^T_m−i

0(σ1,σ)⊗In)u1(λ), . . . ,(e^T_m−i

0(σ1,σ)⊗In)up(λ)

)

is a left minimal basis ofG(λ).

Further, ifη₁ ≤ · · · ≤η_p are the left minimal indices ofL(λ)thenη₁−c_L≤ · · · ≤η_p−c_L are the left minimal indices of G(λ).

Similarly, we have the following recovery rules for eigenvectors of G(λ) from those of the FPRs of G(λ).

Theorem 6.3.6. Suppose that G(λ)is regular and µ∈C is an eigenvalue of G(λ). Let L(λ) := M^Sτ1M^Sσ1(λM^Sτ −M^Sσ)M^Sσ2M^Sτ2 be an FPR of G(λ). Let w_i :=



 ui

v_i



 ∈ C^mn+r, where u_i ∈C^mn and v_i ∈C^r for i= 1 :p. Then we have the following.

(a) Right eigenvectors. If (w₁, . . . , w_p) is a basis of N_r(L(µ))then

(

(e^T_m−c

0(σ,σ2)⊗ I_n)u₁, . . . ,(e^T_m−c

0(σ,σ2)⊗I_n)u_p

)

is a basis of N_r(G(µ)).

(b) Left eigenvectors. If (w1, . . . , wp) is a basis of Nl(L(µ)) then

(

(e^T_m−i

0(σ1,σ) ⊗ I_n)u₁, . . . ,(e^T_m−i

0(σ1,σ)⊗I_n)u_p

)

is a basis of N_l(G(µ)).

Proof. Since L(λ) is a linearization of S(λ), we have eig(S) = eig(L).Again since S(λ) is irreducible, we have eig(G) ⊂ eig(S). Hence as a consequence of Theorem 6.3.4 and Theorem 5.5.8, the desired results follow.

We now illustrate eigenvector recovery rule for G(λ) by considering an example Example 6.3.7. Consider G(λ) := P4

i=0λⁱA_i + C(λE − A)⁻¹B. Now choose σ :=

(1,2,3,0), σ₁ := ∅, σ₂ := (2,1), τ := (−4), τ₁ = ∅ and τ₂ := ∅. The FPR of G(λ) associated with (σ₁, σ, σ₂) and (τ₁, τ, τ₂) is given by L(λ) := λM^S−4−M^S(1,2,3,0)

M^S(2,1)=























λA₄+A₃ A₂ A₁ −I_n 0 A₂ −λA₂−I_n −λA₁ λI_n 0

A₁ λI_n A₀ 0 C

−I_n 0 λI_n 0 0

0 0 B 0 A−λE























Let



 u v



∈ Nr(L(λ)) and



 x y



∈ Nl(L(λ)) be such that {u, x} ⊂ C⁴ⁿ and {v, y} ⊂C^r. We have c0(σ, σ2) = 1 and i0(σ1, σ) = 1. Hence by Theorem 6.3.4,





(e^T₄₋₁⊗I_n)u v



 =





(e^T₃ ⊗I_n)u v



 ∈ N_r(S(λ)) and





(e^T₃ ⊗I_n)x y



 ∈ N_l(S(λ)). To verify the recovery rule, define u_i := (e^T_i ⊗I_n)u and x_i := (e^T_i ⊗I_n)x, for i= 1 : 4, and consider L(λ)



 u v



= 0.

This gives

(λA₄ +A₃)u₁+A₂u₂+A₁u₃−u₄ = 0 (6.8) A2u1+ (−λA2−In)u2−λA1u3+λu4 = 0 (6.9) A₁u₁+λu₂+A₀u₃+Cv = 0 (6.10)

−u₁+λu₃ = 0 (6.11)

Bu₃ + (A−λE)v = 0 (6.12)

From (6.11) we have u1 =λu3. Now adding λ times (6.8) with (6.9) we have (λ²A4+ λA₃+A₂)u₁−u₂ = 0⇒u₂ = (λ²A₄+λA₃+A₂)u₁ = (λ³A₄+λ²A₃+λA₂)u₃. Substituting the values of u₁ and u₂ in (6.10), we have (λ⁴A₄+λ³A₃+λ²A₂+λA₁+A₀)u₃+Cv = 0⇒P(λ)u3+Cv = 0.Hence by (6.12), we haveS(λ)



 u₃

v



= 0, i.e.,



 u₃

v



∈ Nr(S(λ)).

Similarly, we can verify that



 x₃

y



 ∈ N_l(S(λ)). It is easily seen that G(λ)u₃ = 0 and x^T₃G(λ) = 0.

7

Structured Strong Linearizations of Structured Rational Matrices

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew- Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian ra- tional matrices arise in many applications. For structured rational matrices, it is desir- able to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices. The primary aim of this chapter is to construct structure-preserving Rosenbrock strong linearizations of structured (sym- metric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian) rational matrices. For this purpose, we pro- pose an infinite family of Fiedler-like pencils (which we refer to as generalized Fiedler pencils with repetition (GFPRs)) and show that the family of GFPRs is a rich source of structure-preserving strong linearizations of structured rational matrices. We con- struct symmetric, skew-symmetric, Hamiltonian, skew-Hamiltonian, Hermitian, skew- Hermitian, para-Hermitian and para-skew-Hermitian strong linearizations of a rational matrixG(λ) whenG(λ) has the same structure. Further, whenG(λ) is real and symmet- ric, we show that the real symmetric linearizations ofG(λ) preserve the Cauchy-Maslov index of G(λ). We describe the recovery of eigenvectors, minimal bases and minimal indices of G(λ) from those of the linearizations of G(λ) and show that the recovery is operation-free. We also show that FPs, GFPs and FPRs of G(λ) constructed in Chapter 6 are Rosenbrock strong linearizations of G(λ).

7.1 Introduction

Structured rational matrices such as symmetric, skew-symmetric, Hamiltonian, skew- Hamiltonian, Hermitian, skew-Hermitian, para-Hermitian and para-skew-Hermitian ra- tional matrices arise in many applications, see [38, 41, 36, 32, 49, 55, 64, 35] and the references therein. For example, the Hermitian rational eigenvalue problem

G(λ)u:=

λ²M +K −

k

X

i=1

1

1 +λb_i∆K_i u= 0

arises in the study of damped vibration of a structure, where M and K are positive definite, b_i is a relaxation parameter and ∆K_i is an assemblage of element stiffness matrices [49, 55]. Also various structured rational matrices arise as transfer functions of linear time-invariant (LTI) systems, see [38, 41, 36, 32, 50, 64]. For structured rational matrices, it is desirable to construct structure-preserving linearizations so as to preserve the symmetry in the eigenvalues and poles of the rational matrices.

Our main aim in this chapter is to construct structure-preserving strong lineariza- tions of structured rational matrices and to recover eigenvectors, minimal bases and minimal indices of rational matrices from those of the linearizations. We consider the following structures:

symmetric : G(λ)^T =G(λ) Hermitian : G(λ)^∗=G(¯λ) skew-symmetric : G(λ)^T =−G(λ) skew-Hermitian : G(λ)^∗=−G(¯λ) Hamiltonian : G(λ)^T =G(−λ) para-Hermitian : G(λ)^∗=G(−¯λ) skew-Hamiltonian :G(λ)^T =−G(−λ) para-skew-Hermitian :G(λ)^∗=−G(−λ),¯

(7.1)

whereX^T (resp., X^∗) denotes the transpose (resp., conjugate transpose) of a matrix X and ¯λ denotes the conjugate of λ. For more on these structured rational matrices, we refer to [38, 41, 36, 49, 50, 64, 32, 55] and the references therein.

The Cauchy-Maslov index [23] (also known as the matrix Cauchy index [10]) of a real symmetric rational matrix plays an important role in applications such as in networks of linear systems, see [22, 39] and the references therein. If G(λ) is real symmetric then the Cauchy-Maslov index of G(λ) is defined as Ind_CM(G) := (# eigenvalues of G(λ) which jump from−∞to +∞) − (# eigenvalues ofG(λ) which jump from +∞to−∞) as the real parameterλ traverses from −∞ to +∞, see [10]. It is therefore desirable to construct real symmetric linearizations of G(λ) that preserve the Cauchy-Maslov index of G(λ).

We mention that there is a slight difference in the naming convention between some of the structured rational matrices and structured matrix polynomials. The Hamiltonian

(resp., skew-Hamiltonian) structure for rational matrices is known as T-even (resp.,T- odd) structure for matrix polynomials [45]. On the other hand, para-Hermitian (resp., para-skew-Hermitian) structure for rational matrices is known as ∗-even (rep., ∗-odd) structure for matrix polynomials [45]. We follow both the naming conventions in the rest of the chapter without any bias.