GFPs of rational matrices

which implies Ord(Ce) = Ord(B)e −p i(σ) ≥ Ord(B)−p i(σ) = Ord(C). Hence C is a minimal basis ofNr(S). This proves (a).

By part (a), we have that C is a right minimal basis of S(λ) and deg(x_j(λ)) = deg Fσ(S)w_j(λ)

= deg(w_j(λ))−i(σ) for j = 1 :p. This shows that ifε₁ ≤ · · · ≤ε_p are the right minimal indices ofLσ(λ) thenε₁−i(σ)≤ · · · ≤ε_p−i(σ) are the right minimal indices of S(λ). This proves (b).

The results for left minimal bases and left minimal indices, respectively, follow from right minimal bases and right minimal indices in view of the following facts. First, x(λ) ∈ N_l(Lσ) ⇐⇒ x(λ) ∈ N_r(L^Tσ). Second, (M^Sj)^T’s are Fiedler matrices of S(λ)^T. Third, if Lσ(λ) is the Fiedler pencil of S(λ) associated with a permutation σ then L^σ(λ)^T is the Fiedler pencil of S(λ)^T associated with the permutation rev(σ). Fourth, CIP(σ) = (c0, i0) ⇐⇒ CIP(rev(σ)) = (i0, c0). Fifth, we have i(σ) = c(rev(σ)). This completes the proof.

As a consequence of Theorem 6.1.13 and Theorem 5.5.8 we have the following result.

Theorem 6.1.14. LetLσ(λ)be a Fiedler pencil ofS(λ)associated with a permutationσ of {0,1, . . . , m−1}. Let c(σ)and i(σ), respectively, be the total number of consecutions and inversions of σ. Suppose that CIP(σ) = (c₀, i₀). Let w_i(λ) :=



 u_i(λ) v_i(λ)



 ∈C[λ]^mn+r, where u_i(λ)∈C[λ]^mn and v_i(λ)∈C[λ]^r for i= 1 :p.

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

is a right minimal basis ofLσ(λ) then

(

(e^T_m−c0⊗I_n)u₁(λ), . . . ,(e^T_m−c0⊗I_n)u_p(λ)

)

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

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

is a left minimal basis of L^σ(λ) then

(

(e^T_m−i0⊗In)u1(λ), . . . ,(e^T_m−i0⊗In)up(λ)

)

is a left minimal basis ofG(λ). Further, if η₁ ≤ · · · ≤η_p are the left minimal indices of L^σ(λ) then η₁−c(σ) ≤ · · · ≤ η_p −c(σ) are the left minimal indices of G(λ).

and M^S±i fori= 0 : m, associated with P(λ) and G(λ), respectively, as given in Section 6.1.

A permutation w := (w₀, w₁) of {0,1, . . . , m} is said to be proper if 0 ∈ w₀ and m ∈w₁, otherwise w is said to be non-proper.

Definition 6.2.1 (PGF pencil, [4]). Let ω:= (ω₀, ω₁)be a permutation of {0,1, . . . , m}.

Define Tω(λ) := λM^S−ω1 −M^Sω0. If ω is a proper permutation, then Tω(λ) is called a proper generalized Fiedler (PGF) pencil of G(λ)(or PGF pencil ofS(λ)) associated with ω. Otherwise Tω(λ) is called a non-proper generalized Fiedler (NPGF) pencil of G(λ) (or NPGF pencil of S(λ)) associated with ω. Moreover, Tω(λ) is said to be a Type-I NPGF pencil of G(λ) if 0∈w₀ and m∈w₀.

Remark 6.2.2. Note that (M^S−m)⁻¹ does not exist when S(λ) is singular and hence in such a case Type-I NPGF pencil is not defined. Also, since computation of(M^S0)⁻¹ would involve a substantial amount of work, for simplicity, we consider only PGF pencils of G(λ).

Let Tω(λ) := λM^S−ω1 −M^Sω0 be the PGF pencil of G(λ) associated with a per- mutation ω := (ω₀, ω₁) of {0 : m}. Let ω₁ be given by ω₁ := (σ₁, m, σ₂). Set σ :=

(rev(σ₁), ω₀, rev(σ₂)).Thenσ is a permutation of{0 :m−1}.Consider the Fiedler pen- cilLσ(λ) := λM^S−m−M^Sσ ofG(λ) associated withσ.ThenLσ(λ) = M^Srev(σ1)Tω(λ)M^Srev(σ2), whereM^Srev(σ1)andM^Srev(σ2)are nonsingular. This implies that any PGF pencil is strictly equivalent to a Fiedler pencil. Hence the left (resp., right) minimal indices of T^ω(λ) and the left (resp., right) minimal indices of L^σ(λ) are the same. Moreover, any PGF pencil T^ω(λ) is strictly equivalent to a Fiedler pencil that preserves the consecution of ω₀ at 0 (see Theorem 6.2.3) except for the following particular case:

Tδ(λ) :=λM^S−mM^S−(m−1)· · ·M^S−(c0+1)−M^S0M^S1 · · ·M^Sc0 =λM^S−δ1 −M^Sδ0, (6.6) where c₀ ∈ {0,1, . . . , m−2}. In (6.6) the permutation δ:= (δ₀, δ₁) is defined by

δ₀ := (0,1, . . . , c₀) and δ₁ := (m, m−1, . . . , c₀+ 1). (6.7) Theorem 6.2.3. [4] Let T^ω(λ) :=λM^S−ω₁ −M^Sω0 be the PGF pencil of G(λ) associated with a permutation ω := (ω₀, ω₁) of {0,1, . . . , m}. Suppose that CIP(ω₀) = (c₀, i₀) and ω₁ 6= (m, m−1, . . . , c₀ + 1). Let ω₁ be given by ω₁ := (σ₁, m, σ₂). If c₀+ 1 ∈ ω₀ ∪σ₁ then define ξ₁ := rev(σ₁) and ξ₂ := rev(σ₂). On the other hand, if c₀ + 1 ∈ σ₂ and σ₂ has p consecutive inversions at c₀+ 1 then setτ₁ := (c₀+p+ 1, c₀+p, . . . , c₀+ 1).Then M^Sσ2 =M^S(τ1,τ2) for some sub-permutation τ₂. Define ξ₁ :=rev(σ₁, τ₁) and ξ₂ :=rev(τ₂).

Then ξ₁ and ξ₂ are sub-permutations of {1,2, . . . , m−1} such that the following hold.

(a) c0 ∈/ ξ2 and c0+ 1 ∈/ ξ2.

(b) σ := (ξ₁, ω₀, ξ₂) is a permutation of {0,1, . . . , m−1} and has c₀ consecutions at 0, that is, σ and ω₀ have the same number of consecutions at 0.

(c) The Fiedler pencil Lσ(λ) := λM^S−m−M^Sσ of G(λ) associated with σ is such that Lσ(λ) = M^Sξ1Tω(λ)M^Sξ2 and that M^Sξi = diag(M_ξ^P

i, I_r), i= 1,2, are nonsingular.

Now we describe the recovery of minimal bases and minimal indices of S(λ) from those of the PGF pencils ofS(λ).

Theorem 6.2.4. Let Tω(λ) := λM^S−ω1 −M^Sω0 be the PGF pencil of S(λ) associated 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)), respectively, be the total number of inversions and con- secutions of the permutation (rev(σ₁), ω₀, rev(σ₂)) of {0 :m−1}.

(I) Right minimal bases and right minimal indices.

(a) F^PGFω (S) : N_r(T^ω) → N_r(S),



 u(λ) v(λ)



 7→





(e^T_m−c0⊗I_n)u(λ) v(λ)



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

is a right minimal basis of T^ω(λ) then F^PGFω (S)w1(λ), . . . ,F^PGFω (S)wp(λ)

is a right minimal basis of S(λ).

(b) Ifε₁ ≤ · · · ≤ε_p are the right minimal indices ofT^ω(λ)thenε₁−i_T ≤ · · · ≤ε_p−i_T are the right minimal indices of S(λ).

(II) Left minimal bases and left minimal indices.

(c) K^PGFω (S) : Nl(T^ω) → Nl(S),



 u(λ) v(λ)



 7→





(e^T_m−i0 ⊗I_n)u(λ) v(λ)



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

is a left minimal basis of T^ω(λ) then K^PGFω (S)w1(λ), . . . ,K^PGFω (S)wp(λ)

is a left minimal basis of S(λ).

(d) If η1 ≤ · · · ≤ηp are the left minimal indices of T^ω(λ)then η1−c_T ≤ · · · ≤ηp−c_T are the left minimal indices of S(λ).

Proof. Any PGF pencil is strictly equivalent to a Fiedler pencil. Thus, there exist sub-permutations ξ₁ and ξ₂ of {1, . . . , m − 1} such that Lσ(λ) := M^Sξ1Tω(λ)M^Sξ2 = λM^S−m−M^Sσ is a Fiedler pencil ofS(λ) associated with the permutation σ:= (ξ₁, ω₀, ξ₂) of {0,1, . . . , m−1}. Since M^Sξ2 is nonsingular, it is easily seen that the map (M^Sξ2)⁻¹ : N_r(Tω) → N_r(Lσ), x(λ)7→ M^Sξ2

−1

x(λ), is an isomorphism and maps a minimal basis ofN_r(Tω) to a minimal basis of N_r(Lσ). On the other hand, by Theorem 6.1.13,Fσ(S) :

Nr(L^σ)→ Nr(S) is an isomorphism and maps a minimal basis of Nr(L^σ) to a minimal basis of Nr(S). Hence F^σ(S)(M^Sξ2)⁻¹ :Nr(T^ω)→ Nr(S) is an isomorphism and maps a minimal basis of N_r(T^ω) to a minimal basis of N_r(S). Note that M^Sξ2 = diag(M_ξ2, I_r).

Case-1: If ω₁ 6= (m, m−1, . . . , c₀ + 1) then by Theorem 6.2.3, we can choose ξ₁ and ξ₂ such that σ preserves the consecution of ω₀ at 0, i.e., σ has c₀ consecutive consecutions at 0. Hence if j ∈ξ₂ then j ≥c₀+ 2. ThusM_ξ^P

2 = diag(∗, I_(c0+1)n). Hence M_ξ⁻¹

2 = diag(∗, I_(c0+1)n), which shows that (e^T_m−c0 ⊗I_n)(M_ξ^P

2)⁻¹ = e^T_m−c0 ⊗I_n. Since σ has c₀ consecutive consecutions at 0, by Theorem 6.1.13, Fσ(S) =





e^T_m−c0 ⊗In

I_r



. Consequently, we have

Fσ(S)(M^Sξ2)⁻¹ =





(e^T_m−c0 ⊗In)(M_ξ^P₂)⁻¹ Ir



=





e^T_m−c0 ⊗In

Ir



=F^PGFω (S).

Since i_T = i(σ), the total number of inversions of σ, the desired results for right minimal bases and right minimal indices follow from Theorem 6.1.13.

Case-2: Suppose that ω1 = (m, m−1, . . . , c0 + 1). Then by choosing ξ1 = ∅ and ξ₂ = (c₀ + 1, . . . , m−1), we have σ := (ξ₁, ω₀, ξ₂) = (0,1, . . . , m−1) and the Fiedler pencil L^σ(λ) =T^ω(λ)M^Sξ2. Hence σ has m−1 consecutive consecutions at 0. Thus by Theorem 6.1.13,Fσ(S) =





e^T_m−(m−1)⊗In

I_r



.By (2.12), we have (e^T_m−j⊗I_n)(M_k^P)⁻¹ = e^T_m−(j−1)⊗Infork =j,j = 1 :m−1.Hence it is easy to see that (e^T_m−(m−1)⊗In)(M_ξ^P₂)⁻¹ = (e^T_m−(m−1)⊗In)(M_m−1^P )⁻¹(M_m−2^P )⁻¹· · ·(M_c^P₀₊₁)⁻¹ =e^T_m−c0 ⊗In. Consequently, we have

Fσ(S)(M^Sξ2)⁻¹ =





(e^T_m−(m−1)⊗In)(M_ξ^P₂)⁻¹ Ir



=





e^T_m−c0 ⊗In

Ir



=F^PGFω (S).

Again, since i_T = m −1 = i(σ), the total number of inversions of σ, the desired results for right minimal bases and right minimal indices follow from Theorem 6.1.13.

The proof is similar for left minimal bases and left minimal indices in view of the following facts. First, x(λ) ∈ N_l(T^ω) ⇐⇒ x(λ) ∈ N_r(T^Tω). Second, (M^Sj)^T’s are Fiedler matrices of S(λ)^T. Third, if T^ω(λ) is the PGF pencil of S(λ) associated with a permutation ω := (ω₀, ω₁) then Tω(λ)^T is the PGF pencil of S(λ)^T associ- ated with the permutation ωb := (rev(ω₀), rev(ω₁)). Fourth, CIP(ω₀) = (c₀, i₀) ⇐⇒

CIP(rev(ω₀)) = (i₀, c₀). Fifth, i_T = c_T^T, where c_T^T := c(σ₂, rev(ω₀), σ₁). Indeed, we have c(σ₂, rev(ω₀), σ₁) = i rev((σ₂, rev(ω₀), σ₁))

= i(rev(σ₁), ω₀, rev(σ₂)) = i_T. This completes the proof.

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).