Algorithms for FPRs and GFPRs

A GFPR can be constructed explicitly without multiplying the Fiedler and elementary matrices associated withP(λ), that is, one can easily construct a GFPR only by looking at the index tuples and the matrix assignments those are used to define the GFPR. In this section, we present algorithms which construct FPRs and GFPRs. For this purpose, we define some technical terms which will be useful to describe the algorithms.

Definition 2.4.1. Let α be an index tuple containing indices from {0 : d}, for some integer d≥0, such that α satisfies the SIP. Suppose thatt ∈α. Let thersf(α)be given by rsf(α) =

(

rev(a₀ : 0), rev(a₁ : 1), . . . , rev(a_k : k), rev(a_k+1 : k+ 1). . . , rev(a_d: d)

)

. Then we say that α has p ( ≥ 0) reverse consecutions at t if t ∈ rev(a_k : k) and a_k = t−p for smallest the integer k. We denote the number of reverse consecution of α at t by rct(α). Further, we define α^R(t) := rev(ak+1 :k+ 1), . . . , rev(ad :d)

, where t ∈rev(ak :k) for the smallest integer k.

Example 2.4.2. Let α = (3,1,0,2,4,1,3). Then rsf(α) = (1,0,3,2,1,4,3) = (rev(0 : 1), rev(1 : 3), rev(3 : 4)) =: rev(a₀ : 0), rev(a₁ : 1), rev(a₂ : 2), rev(a₃ : 3), rev(a₄ : 4)

,

where a1 = 0, a3 = 1, a4 = 3 and a0 = ∞ = a2. Then we have rc2(α) = 1 since 2 ∈ rev(a3 : 3) and a3 = 2−1. Further, we have α^R(2) = rev(a4 : 4) = rev(3 : 4).

Similarly, we have rc₁(α) = 1 and α^R(1) = (rev(1 : 3), rev(3 : 4)).

The reverse consecutions for index tuple containing negative indices is defined as follows.

Definition 2.4.3. Letβ be an index tuple containing indices from {−d:−1}, for some integer d ≥ 1, such that β satisfies the SIP. Suppose that −t ∈ β. Let the rsf(β) be given by

rsf(β) =

rev(−ad:−d), . . . , rev(−ak+1 :−(k+1)), rev(−ak:−k), . . . , rev(−a1 :−1) . Then we say that β has q ( ≥0) reverse consecutions at −t if −t ∈rev(−a_k :−k) and

−a_k =−t−qfor the largest integer k. We denote the number of reverse consecution of β at −t by rc−t(β). Further, we define β^R(−t) := (rev(−ak−1 : −(k−1)), . . . , rev(−a₁ :

−1)), where −t∈rev(−a_k :−k) for the largest integer k.

Definition 2.4.4. Let α be an index tuple containing indices from {−m :m−1}. Sup- pose that s∈α. Then the position of the first occurrence of s in α is denoted by p_s(α).

For example, let α = (0,1,2,3,2,−1,−5,4,3,−1) be an index tuple and X = (X₁, X₂, . . . , X₁₀) be an arbitrary matrix assignment forα.Thenp₂(α) = 3 andp−1(α) = 6 as the first occurrence of 2 and−1 appear in 3rdand 6th positions in α, respectively.

The following Algorithm-1 and Algorithm-2 construct an FPR ofP(λ), and Algorithm- 3 and Algorithm-4 construct a GFPR of P(λ). For proof see Appendix A.

Algorithm 1: λL₁−L₀ :=M_τ^P₁M_σ^P₁(λM_τ^P −M_σ^P)M_σ^P₂M_τ^P₂ be an FPR of P(λ) Input : σ, τ, σ_j and τ_j, j = 1,2.

Output : L₀

Notation: α:=rsf(σ₁, σ, σ₂) and β :=rsf(τ₁, τ₂).

L<sup>`</sup><sub>0</sub> :=`^th block row of L₀ = (e^T_` ⊗I_n)L₀ for i= 0 :h do

if the subtuple of α with indices {i, i+ 1} starts with i+ 1 then L^m−i₀ =e^T_m−(i+1+c

i+1(α))⊗I_n else

if 06=i−rc_i(α) =: p, set α^R:=α^R(i) then L^m−i₀ = (e^T_m−(p+c

p(α^R))⊗I_n)−

i

P

j=p

e^T_m−(j+1+c

j+1(α^R))⊗A_j else

L^m−i₀ =−

i

P

j=0

e^T_m−(j+1+c

j+1(α^R))⊗A_j for i=h+ 1 :m−1 do

if −i,−(i+ 1)∈/ β or the subtuple of β with indices {−i,−(i+ 1)} starts with

−ithen

L^m−i₀ =e^T_{m−(i−1−c}

−i(β))⊗I_n else

if −m6=−(i+rc_−(i+1)(β) + 1) =:−q, set β^R :=β^R(−(i+ 1)) then L^m−i₀ = (e^T_{m−(q−1−c}

−q(β^R))⊗I_n) +

q−1

P

j=i

e^T_{m−(j−1−c}

−j(β^R))⊗A_j+1 else

L^m−i₀ =

m−1

P

j=i

(e^T_{m−(j−1−c}

−j(β^R))⊗A_j+1) returnL0

Algorithm 2: λL₁−L₀ :=M_τ^P₁M_σ^P₁(λM_τ^P −M_σ^P)M_σ^P₂M_τ^P₂ be an FPR of P(λ) Input : σ, τ, σ_j and τ_j j = 1,2.

Output :L₁

Notation:α :=rsf(σ₁, σ₂) and β :=rsf(τ₁, τ, τ₂).

L<sup>`</sup><sub>1</sub> :=`^th block row of L₁ = (e^T_` ⊗I_n)L₁ for i= 0 :h−1 do

if i, i+ 1∈/ α or the subtuple of α with indices {i, i+ 1} starts with i+ 1then L^m−i₁ =e^T_m−(i+1+c

i+1(α))⊗I_n else

if 06=i−rc_i(α) =:p, set α^R:=α^R(i) then L^m−i₁ = (e^T_m−(p+c

p(α^R))⊗I_n)−Pⁱ

j=p

(e^T_m−(j+1+c

j+1(α^R))⊗A_j) else

L^m−i₁ =−

i

P

j=0

(e^T_m−(j+1+c

j+1(α^R))⊗A_j) for i=h:m−1do

if the subtuple of β with indices {−i,−(i+ 1)} starts with −ithen L^m−i₁ =e^T_{m−(i−1−c}

−i(β))⊗I_n else

if −m6=−(i+rc−(i+1)(β) + 1) =:−q, set β^R:=β^R(−(i+ 1)) then L^m−i₁ = (e^T_{m−(q−1−c}

−q(β^R))⊗I_n) +

k−1

P

j=i

(e^T_{m−(j−1−c}

−j(β^R))⊗A_j+1) else

L^m−i₁ =

m−1

P

j=i

(e^T_{m−(j−1−c}

−j(β^R))⊗A_j+1) return L₁

Algorithm 3: λL₁ −L₀ := M_(τ1,σ1)(Y₁, X₁)(λM_τ^P − M_σ^P)M_(σ2,τ2)(X₂, Y₂) be a GFPR of P(λ)

Input : σ, τ, σ_j and τ_j, j = 1,2. Matrix assignments X_j and Y_j, j = 1,2.

Output : L₀

Notation: α:=rsf(σ₁, σ, σ₂), Z :=rsf(X₁, P, X₂), where P is the trivial matrix assignment for σ,β :=rsf(τ₁, τ₂) and W :=rsf(Y₁, Y₂).

L<sup>`</sup><sub>0</sub> :=`^th block row of L₀ = (e^T_` ⊗I_n)L₀. for i= 0 :h do

if the subtuple of α with indices {i, i+ 1} starts with i+ 1 then L^m−i₀ =e^T_m−(i+c

i+1(α)+1)⊗I_n else

if 06=i−rc_i(α) =: k, set α^R:=α^R(i) then L^m−i₀ =e^T_m−(k+c

k(α^R))⊗I_n+

i

P

j=k

e^T_m−(j+1+c

j+1(α^R))⊗Z_pi(α)+i−j else

L^m−i₀ =

i

P

j=0

e^T_m−(j+1+c

j+1(α^R))⊗Z_pi(α)+i−j

for i=h+ 1 :m−1 do

if −i,−(i+ 1)∈/ β or the subtuple of β with indices {−i,−(i+ 1)} starts with

−ithen

L^m−i₀ =e^T_m−(i−c

−i(β)−1)⊗I_n else

if −m6=−(i+rc_−(i+1)(β) + 1) =:−k, set β^R :=β^R(−(i+ 1)) then L^m−i₀ =e^T_m−(k−c

−k(β^R)−1)⊗I_n+

k−1

P

j=i

e^T_m−(j−c

−j(β^R)−1)⊗W_{p−(i+1)(β)+i−j} else

L^m−i₀ =

m−1

P

j=i

e^T_m−(j−c

−j(β^R)−1)⊗W_{p−(i+1)(β)+i−j} returnL₀

Algorithm 4:λL₁−L₀ :=M_(τ1,σ1)(Y₁, X₁)(λM_τ^P−M_σ^P)M_(σ2,τ2)(X₂, Y₂) be a GFPR of P(λ)

Input : σ, τ, σ_j and τ_j, j = 1,2. Matrix assignments X_j and Y_j,j = 1,2.

Output :L₁

Notation:α :=rsf(σ₁, σ₂), Z :=rsf(X₁, X₂),β :=rsf(τ₁, τ, τ₂) and

W :=rsf(Y₁, P, Y₂), where P is the trivial matrix assignment forτ. L<sup>`</sup><sub>1</sub> :=`^th block row of L₁ = (e^T_` ⊗I_n)L₁.

for i= 0 :h−1 do

if i, i+ 1∈/ α or the subtuple of α with indices {i, i+ 1} starts with i+ 1then L^m−i₁ :=e^T_m−(i+c

i+1(α)+1)⊗I_n else

if 06=i−rci(α) =:k, set α^R:=α^R(i) then L^m−i₁ := (e^T_m−(k+c

k(α^R))⊗I_n) +

i

P

j=k

(e^T_m−(j+1+c

j+1(α^R))⊗Z_pi(α)+i−j) else

L^m−i₁ :=

i

P

j=0

(e^T_m−(j+1+c

j+1(α^R))⊗Z_pi(α)+i−j) for i=h:m−1do

if the subtuple of β with indices {−i,−(i+ 1)} starts with −ithen L^m−i₁ :=e^T_m−(i−c

−i(β)−1)⊗I_n else

if −m6=−(i+rc−(i+1)(β) + 1) =:−k, set β^R :=β^R(−(i+ 1)) then L^m−i₁ := (e^T_m−(k−c

−k(β^R)−1)⊗In) +

k−1

P

j=i

(e^T_m−(j−c

−j(β^R)−1)⊗W_p−(i+1)(β)+i−j) else

L^m−i₁ :=

m−1

P

j=i

(e^T_m−(j−c

−j(β^R)−1)⊗W_p−(i+1)(β)+i−j) return L₁

3

EGFPRs of Matrix Polynomials

The basic building blocks of GFPRs ofP(λ) are special GFPs ofP(λ) of the formλM_τ^P− M_σ^P as given in Definition 2.1.9 which in turn are determined by a sub-permutation {0 : h} of {0 : m −1}. In this chapter, we show that by replacing λM_τ^P −M_σ^P by any GFPs of P(λ) results in a large family of pencils which we refer to as extended GFPRs (EGFPRs) of P(λ) and show that the EGFPRs are strong linearizations of P(λ). The EGFPRs have the same distinctive features as Fiedler pencils, namely, the construction of EGFPRs is operation-free and the recovery of eigenvectors, minimal bases and minimal indices of P(λ) from those of the EGFPRs is also operation-free.

Also, most of the structure-preserving strong linearizations that have been constructed in literature are obtained from GFPs and GFPRs. We show that the family of EGFPRs subsumes both GFPs and GFPRs and expands the arena in which to look for structure- preserving strong linearizations of P(λ).

3.1 EGFPRs of matrix polynomials

Recall thatP(λ) :=Pm

j=0λ^jA_j ∈C[λ]^n×n.We consider the elementary matricesM_±i(X) and Fiedler matrices M_±i^P, for i= 0 :m, associated with P(λ) as given in Chapter 2.

It is clear from the definitions of GFPs, FPRs and GFPRs ofP(λ) that not all the GFPs are FPRs, and vice versa. For example, consider the GFP T(λ) =λM_{(−4,−3,−1)}^P − M_(0,2)^P and the FPR L(λ) = (λM_−4,−3^P − M_(0,1,2)^P )M₁^P of P(λ) := P4

i=0λⁱA_i. Then T(λ) is not an FPR and L(λ) is not a GFP. Further, consider the pencil F(λ) :=

(λM_(−4,−1)^P − M_(2,3,0)^P )M₂^P. Then F(λ) is neither a GFP nor an FPR of P(λ). This motivates us to present a unified framework for construction of Fiedler-like pencils of P(λ) which subsumes FPs, GFPs, FPRs, and GFPRs of P(λ).

Definition 3.1.1 (EGFPR and EFPR ofP(λ)). Let(σ, ω)be a permutation of{0 :m}.

Set τ :=−ω. Let σ1 and σ2 be index tuples containing indices from σ\ {m−1, m} such that(σ₁, σ, σ₂)satisfies the SIP. Similarly, letτ₁ andτ₂be index tuples containing indices from τ \ {−1,−0} such that (τ₁, τ, τ₂) satisfies the SIP. Let X₁, X₂, Y₁ and Y₂ be any arbitrary matrix assignments for σ₁, σ₂, τ₁ and τ₂, respectively. Then the pencil

L(λ) := Mτ1(Y1)Mσ1(X1) (λM_τ^P −M_σ^P)Mσ2(X2)Mτ2(Y2) (3.1) is said to be an extended generalized Fiedler pencil with repetition (EGFPR) of P(λ).

In particular, if all the matrix assignments X_j and Y_j, j = 1,2, are the trivial matrix assignments then the resulting pencil L(λ) :=M_τ^P₁M_σ^P₁(λM_τ^P −M_σ^P)M_σ^P₂M_τ^P₂ is said to be an extended Fiedler pencil with repetition (EFPR) of P(λ).

It follows from Definition 3.1.1 that FPs, GFPs, FPRs and GFPRs of P(λ) are subclasses of EGFPRs.

Example 3.1.2. Let P(λ) := P6

i=0λⁱA_i and let σ := (0,1,4,5), σ₁ := ∅, σ₂ :=

(0,4), τ := (−2,−3,−6), τ₁ := (−3) and τ₂ := ∅. Let X and (Y, Z) be any arbi- trary matrix assignments for (−3) and (0,4), respectively. Then the pencil L(λ) :=

M−3(X) λM_{(−2,−3,−6)}^P −M_(0,1,4,5)^P

M(0,4)(Y, Z) =





























λA6+A5 −Z −In 0 0 0

A₄ λZ −I_n λI_n 0 0 0

0 0 0 −In λIn 0

−I_n 0 0 λI_n−X λX 0

0 λIn 0 λA3 λA2+A1 −Y

0 0 0 0 A₀ λY





























is an EGFPR of P(λ).

Remark 3.1.3. It follows from Definition 3.1.1 that (τ₁, σ₁) 6∼ (σ₁, τ₁) and (σ₂, τ₂) 6∼

(τ₂, σ₂) in general. So M_τ1(Y₁)M_σ1(X₁) 6= M_σ1(X₁)M_τ1(Y₁) and M_σ2(X₂)M_τ2(Y₂) 6=

M_τ2(Y₂)M_σ2(X₂). Thus, for L(λ) := M_τ1(Y₁)M_σ1(X₁)(λM_τ^P − M_σ^P)M_σ2(X₂)M_τ2(Y₂) and T(λ) :=Mσ1(X1) Mτ1(Y1)(λM_τ^P −M_σ^P)Mτ2(Y2)Mσ2(X2), we have L(λ)6=T(λ). By interchanging the positions of M_σj(X_j) with M_τj(Y_j) in L(λ) we obtain a new family of pencils for P(λ).

Note that for an index tuple αand a matrix assignment X ofα,M_α(X) is operation free if and only if M_α^P is operation free, see Lemma 2.1.7 and Remark 2.1.8. Also note

that M_α^P is not operation free if m−1 and m simultaneously belong toα, or if −1 and

−0 simultaneously belong to α. The following result shows that a subclass of EGFPRs of P(λ) is always operation free.

Proposition 3.1.4. Let L(λ) := M_τ1(Y₁)M_σ1(X₁)(λM_τ^P − M_σ^P)M_σ2(X₂)M_τ2(Y₂) =:

λL₁ −L₀ be an EGFPR of P(λ) such that m −1 and m simultaneously do not be- long to σ, and −1 and −0 simultaneously do not belong to τ. Then L(λ) is operation free.

Proof. See Appendix B.

The following result shows that, with some generic nonsingularity conditions, the EGFPRs are strong linearization of P(λ).

Theorem 3.1.5. If all the matrix assignments X_j and Y_j, j = 1,2, are nonsingular, then the EGFPR L(λ) as given in Definition 3.1.1 is a strong linearization of P(λ).

Proof. We haveL(λ) =M_(τ1,σ1)(Y₁, X₁)T(λ)M_(σ2,τ2)(X₂, Y₂), whereT(λ) :=λM_τ^P−M_σ^P is a GF pencil of P(λ). As Xj and Yj, j = 1,2, are nonsingular matrix assignments, M_(τ1,σ1)(Y₁, X₁) andM_(σ2,τ2)(X₂, Y₂) are nonsingular. SinceT(λ) is a strong linearization of P(λ), it follows that L(λ) is a strong linearization of P(λ).

Assumption: Henceforth, for simplicity we assume that the matrix assignments are always nonsingular.